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http://20000-names.com/wolf_names.htm
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20000-NAMES.COM: Wolf Names--meaning, origin, etymology
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KENYON : Irish surname transferred to forename use, from an Anglicized form of Gaelic Mac Coinín "son of Coinín ," hence "little wolf." KUCKUNNIWI : Native American Cheyenne name meaning "little wolf." LIULFR : Old Norse name of uncertain etymology, possibly composed of the elements hlíf "shield, protection" and ulfr "wolf," hence "shield wolf." LOPE : Spanish form of Latin Lupus, meaning "wolf." LOUP : French form of Latin Lupus, meaning "wolf."
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http://20000-names.com/wolf_names.htm
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20000-NAMES.COM: Wolf Names--meaning, origin, etymology
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Old Norse name of uncertain etymology, possibly composed of the elements hlíf "shield, protection" and ulfr "wolf," hence "shield wolf." LOPE : Spanish form of Latin Lupus, meaning "wolf." LOUP : French form of Latin Lupus, meaning "wolf." LOUVEL : Old Norman French byname derived from a diminutive form of the word lou "wolf," hence "little wolf." LOVEL : Variant spelling of English Lovell, meaning "little wolf." LOVELL :
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http://20000-names.com/wolf_names.htm
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20000-NAMES.COM: Wolf Names--meaning, origin, etymology
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LOUVEL : Old Norman French byname derived from a diminutive form of the word lou "wolf," hence "little wolf." LOVEL : Variant spelling of English Lovell, meaning "little wolf." LOVELL : English surname transferred to forename use, from a variant spelling of English Lowell , meaning "little wolf." LOWELL : English surname transferred to forename use, derived from the Old Norman French byname Louvel, meaning "little wolf." LUPUS : Latin name derived from the word lupus, meaning "wolf."
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msmarco_v2.1_doc_00_8034857#31_12397841
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http://20000-names.com/wolf_names.htm
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20000-NAMES.COM: Wolf Names--meaning, origin, etymology
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English surname transferred to forename use, from a variant spelling of English Lowell , meaning "little wolf." LOWELL : English surname transferred to forename use, derived from the Old Norman French byname Louvel, meaning "little wolf." LUPUS : Latin name derived from the word lupus, meaning "wolf." LYCAON : Latin form of Greek Lykaon, possibly meaning "wolf." In mythology, this is the name of an early king of Arkadia. LYCURGUS : Latin form of Greek Lykourg
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http://2000clicks.com/MathHelp/BasicFactorialConsecutiveIntegerProducts.aspx
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Product of Consecutive Integers
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Product of Consecutive Integers
Product of Consecutive Integers
Contents of this page
Product of an Odd Number of Consecutive Integers
Product of an Even Number of Consecutive Integers
Ratio of Factorials, and Gosper's Approximation
The product of n consecutive integers is divisible by n!
A note about products of negative numbers
Proof that floor [a+b] ≥ floor [a] + floor [b]
Internet references
Related pages in this website
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Product of Consecutive Integers
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Math Help > Basic Math > Factorials, Consecutive Integer Products > Product of Consecutive Integers
Contents of this page
Who knew there was so much to say about the product of consecutive integers? This page answers the following questions: What are the coefficients of the polynomial giving the product of 2k+1 consecutive integers whose "middle" number is n? What are the coefficients of the polynomial giving the product of 2k consecutive integers whose "middle" numbers add up to n? What is Gosper's approximation to the product of consecutive integers? Why is the product of n consecutive integers divisible by n! ? Product of an Odd Number of Consecutive Integers
The formula for finding the product of any odd number of consecutive integers follows the pattern of coefficients in Sloane's A008955. If the number of numbers is 2k+1, where k=0,1,2,..., then we can define a function P odd (k,n) as the product of 2k+1 numbers in which the "middle" number is n. That is P odd (k,n)= (n-k) (n-k+1)... (n+k)
Notice that P odd (k+1,n) = (n 2 -k 2) P odd (k,n), which gives us a recurrence relation that lets us find the coefficients of each polynomial: P odd (0,n) = (n) = n
P odd (1,n) = (n-1) (n) (n+1) = n 3 -n
P odd (2,n) = (n-2) (n-1)... (n+2) = n 5 -5n 3 +4n
P odd (3,n) = (n-3) (n-2)... (n+3) = n 7 -14n 5 +49n 3 -36n
P odd (4,n) = (n-4) (n-3)... (n+4) = n 9 -30n 7 +273n 5 -820n 3 +576n
P odd (5,n) = (n-5) (n-4)... (n+5) = n 11 -55n 9 +1023n 7 -7645n 5 +21076n 3 -14400n
P odd (6,n) = (n-6) (n-5)... (n+6) = n 13 -91n 11 +3003n 9 -44473n 7 +296296n 5 -773136n 3 +518400n
P odd (7,n) = (n-7) (n-6)... (n+7) = n 15 -... well, you get the idea! Product of an Even Number of Consecutive Integers
The formula for finding the product of any even number of consecutive integers follows the pattern of coefficients in Sloane's A008956.
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Product of Consecutive Integers
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Product of Consecutive Integers
Product of Consecutive Integers
Contents of this page
Product of an Odd Number of Consecutive Integers
Product of an Even Number of Consecutive Integers
Ratio of Factorials, and Gosper's Approximation
The product of n consecutive integers is divisible by n!
A note about products of negative numbers
Proof that floor [a+b] ≥ floor [a] + floor [b]
Internet references
Related pages in this website
|
Why is the product of n consecutive integers divisible by n! ? Product of an Odd Number of Consecutive Integers
The formula for finding the product of any odd number of consecutive integers follows the pattern of coefficients in Sloane's A008955. If the number of numbers is 2k+1, where k=0,1,2,..., then we can define a function P odd (k,n) as the product of 2k+1 numbers in which the "middle" number is n. That is P odd (k,n)= (n-k) (n-k+1)... (n+k)
Notice that P odd (k+1,n) = (n 2 -k 2) P odd (k,n), which gives us a recurrence relation that lets us find the coefficients of each polynomial: P odd (0,n) = (n) = n
P odd (1,n) = (n-1) (n) (n+1) = n 3 -n
P odd (2,n) = (n-2) (n-1)... (n+2) = n 5 -5n 3 +4n
P odd (3,n) = (n-3) (n-2)... (n+3) = n 7 -14n 5 +49n 3 -36n
P odd (4,n) = (n-4) (n-3)... (n+4) = n 9 -30n 7 +273n 5 -820n 3 +576n
P odd (5,n) = (n-5) (n-4)... (n+5) = n 11 -55n 9 +1023n 7 -7645n 5 +21076n 3 -14400n
P odd (6,n) = (n-6) (n-5)... (n+6) = n 13 -91n 11 +3003n 9 -44473n 7 +296296n 5 -773136n 3 +518400n
P odd (7,n) = (n-7) (n-6)... (n+7) = n 15 -... well, you get the idea! Product of an Even Number of Consecutive Integers
The formula for finding the product of any even number of consecutive integers follows the pattern of coefficients in Sloane's A008956. If the number of numbers is 2k, where k=0,1,2,..., then we can define a function P even (k,n) as the product of 2k numbers in which the sum of the two "middle" numbers is n. Notice that this sum is always odd. Each of the consecutive numbers can be represented as (n+m)/2, where m is an odd number that ranges from -2k+1 to 2k-1. That is,
P even (k,n)= (n-2k+1)/2 (n-2k+3)/2 ... (n+2k-1)/2, or
P even (k,n)= (n-2k+1) (n-2k+3) ... (n+2k-1)/2 2k
Notice that P even (k+1,n) = (n 2 - (2k-1) 2) P even (k,n), which gives us a recurrence relation that lets us find the coefficients of each polynomial: P even (0,n) = (1)/2 0 = 1
P even (1,n) = (n-1) (n+1)/2 2 = (n 2 -1)/4
P even (2,n) = (n-3) (n-1) (n+1) (n+3)/2 4 = (n 4 -10n 2 +9)/16
P even (3,n) = (n-5) (n-3)... (n+5)/2 6 = (n 6 -35n 4 +259n 2 -225)/64
P even (4,n) = (n-7) (n-5)... (n+7)/2 8 = (n 8 -84n 6 +1974n 4 -12916n 2 +11025)/256
P even (5,n) = (n-9) (n-7)... (n+9)/2 10 = (n 10 -165n 8 +8778n 6 -172810n 4 +1057221n 2 -893025)/1024
P even (6,n) = (n-11) (n-9)... (n+11)/2 12 = (n 12 -286n 10 +28743n 8 -1234948n 6 +21967231n 4 -128816766n 2 +108056025)/4096
P even (7,n) = (n-13) (n-11)... (n+13)/2 14 = (n 14 -455n 12 +77077n 10 -6092515n 8 +230673443n 6 -3841278805n 4 +21878089479n 2 -18261468225)/16384
Ratio of Factorials, and Gosper's Approximation
Of course, the product of integers from k+1 to n is n!/k!. Toward the end of Mathworld's article in Stirling's Approximation to n!,
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http://2000clicks.com/MathHelp/BasicFactorialConsecutiveIntegerProducts.aspx
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Product of Consecutive Integers
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Product of Consecutive Integers
Product of Consecutive Integers
Contents of this page
Product of an Odd Number of Consecutive Integers
Product of an Even Number of Consecutive Integers
Ratio of Factorials, and Gosper's Approximation
The product of n consecutive integers is divisible by n!
A note about products of negative numbers
Proof that floor [a+b] ≥ floor [a] + floor [b]
Internet references
Related pages in this website
|
If the number of numbers is 2k, where k=0,1,2,..., then we can define a function P even (k,n) as the product of 2k numbers in which the sum of the two "middle" numbers is n. Notice that this sum is always odd. Each of the consecutive numbers can be represented as (n+m)/2, where m is an odd number that ranges from -2k+1 to 2k-1. That is,
P even (k,n)= (n-2k+1)/2 (n-2k+3)/2 ... (n+2k-1)/2, or
P even (k,n)= (n-2k+1) (n-2k+3) ... (n+2k-1)/2 2k
Notice that P even (k+1,n) = (n 2 - (2k-1) 2) P even (k,n), which gives us a recurrence relation that lets us find the coefficients of each polynomial: P even (0,n) = (1)/2 0 = 1
P even (1,n) = (n-1) (n+1)/2 2 = (n 2 -1)/4
P even (2,n) = (n-3) (n-1) (n+1) (n+3)/2 4 = (n 4 -10n 2 +9)/16
P even (3,n) = (n-5) (n-3)... (n+5)/2 6 = (n 6 -35n 4 +259n 2 -225)/64
P even (4,n) = (n-7) (n-5)... (n+7)/2 8 = (n 8 -84n 6 +1974n 4 -12916n 2 +11025)/256
P even (5,n) = (n-9) (n-7)... (n+9)/2 10 = (n 10 -165n 8 +8778n 6 -172810n 4 +1057221n 2 -893025)/1024
P even (6,n) = (n-11) (n-9)... (n+11)/2 12 = (n 12 -286n 10 +28743n 8 -1234948n 6 +21967231n 4 -128816766n 2 +108056025)/4096
P even (7,n) = (n-13) (n-11)... (n+13)/2 14 = (n 14 -455n 12 +77077n 10 -6092515n 8 +230673443n 6 -3841278805n 4 +21878089479n 2 -18261468225)/16384
Ratio of Factorials, and Gosper's Approximation
Of course, the product of integers from k+1 to n is n!/k!. Toward the end of Mathworld's article in Stirling's Approximation to n!, this statement appears: Gosper has noted that a better approximation to n! ( i.e., one which approximates the terms in Stirling's series instead of truncating them) is given by
n! ≈ sqrt ( (2n+1/3)π) n n e -n
Using this approximation, n!/k! is given by
n!/k!
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Product of Consecutive Integers
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Product of Consecutive Integers
Product of Consecutive Integers
Contents of this page
Product of an Odd Number of Consecutive Integers
Product of an Even Number of Consecutive Integers
Ratio of Factorials, and Gosper's Approximation
The product of n consecutive integers is divisible by n!
A note about products of negative numbers
Proof that floor [a+b] ≥ floor [a] + floor [b]
Internet references
Related pages in this website
|
this statement appears: Gosper has noted that a better approximation to n! ( i.e., one which approximates the terms in Stirling's series instead of truncating them) is given by
n! ≈ sqrt ( (2n+1/3)π) n n e -n
Using this approximation, n!/k! is given by
n!/k! ≈ (n/e) n-k (n/k) k sqrt (6n+1) / sqrt (6k+1)
This form is useful for computation, especially when k is almost as large as n, because it avoids very large or very small numbers in intermediate results. The product of n consecutive integers is divisible by n! Proof 1: Consider the number of n-element sets of an (n+m) element set. This is C (n+m,n), which is an integer, and
C (n+m,n) = (m+1) (m+2) (m+3)... (m+n) / n!.
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http://2000clicks.com/MathHelp/BasicFactorialConsecutiveIntegerProducts.aspx
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Product of Consecutive Integers
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Product of Consecutive Integers
Product of Consecutive Integers
Contents of this page
Product of an Odd Number of Consecutive Integers
Product of an Even Number of Consecutive Integers
Ratio of Factorials, and Gosper's Approximation
The product of n consecutive integers is divisible by n!
A note about products of negative numbers
Proof that floor [a+b] ≥ floor [a] + floor [b]
Internet references
Related pages in this website
|
≈ (n/e) n-k (n/k) k sqrt (6n+1) / sqrt (6k+1)
This form is useful for computation, especially when k is almost as large as n, because it avoids very large or very small numbers in intermediate results. The product of n consecutive integers is divisible by n! Proof 1: Consider the number of n-element sets of an (n+m) element set. This is C (n+m,n), which is an integer, and
C (n+m,n) = (m+1) (m+2) (m+3)... (m+n) / n!. The numerator is the product of n consecutive integers, and the denominator is n!, proving that the product of n consecutive integers is divisible by n!. Proof 2: Consider the largest power of each prime factor of the product of consecutive integers. Let P be the product,
P = (m+1) (m+2)... (m+n) = (m+n)!/m!
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Product of Consecutive Integers
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Product of Consecutive Integers
Product of Consecutive Integers
Contents of this page
Product of an Odd Number of Consecutive Integers
Product of an Even Number of Consecutive Integers
Ratio of Factorials, and Gosper's Approximation
The product of n consecutive integers is divisible by n!
A note about products of negative numbers
Proof that floor [a+b] ≥ floor [a] + floor [b]
Internet references
Related pages in this website
|
The numerator is the product of n consecutive integers, and the denominator is n!, proving that the product of n consecutive integers is divisible by n!. Proof 2: Consider the largest power of each prime factor of the product of consecutive integers. Let P be the product,
P = (m+1) (m+2)... (m+n) = (m+n)!/m! For any given positive integer n and prime p, define the function E (n,p) = e such that p e is the largest power of p that divides n!. This is the number of numbers from 1 to n divisible by p plus the number of numbers divisible by p 2 plus the number of numbers divisible by p 3 , etc. E (n,p) = floor [n/p] + floor [n/p 2] + floor [n/p 3] + ...
Using this definition with the product, P= (m+n)!/m!, we see that
E (m+n,p)-E (m,p) = floor [ (m+n)/p] - floor [m/p] + floor [ (m+n)/p 2] - floor [m/p 2] + floor [ (m+n)/p 3] - floor [m/p 3] + ...
which is at least as large as E (n,p) for any given m, n and p, because for any real numbers, a and b, floor [a+b] ≥ floor [a] + floor [b]
A note about products of negative numbers
The two proofs, above work for products of n positive integers, but the statement is also true when the product includes negative integers or zero. There are two cases that need to be considered:
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http://2000clicks.com/MathHelp/BasicFactorialConsecutiveIntegerProducts.aspx
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Product of Consecutive Integers
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Product of Consecutive Integers
Product of Consecutive Integers
Contents of this page
Product of an Odd Number of Consecutive Integers
Product of an Even Number of Consecutive Integers
Ratio of Factorials, and Gosper's Approximation
The product of n consecutive integers is divisible by n!
A note about products of negative numbers
Proof that floor [a+b] ≥ floor [a] + floor [b]
Internet references
Related pages in this website
|
For any given positive integer n and prime p, define the function E (n,p) = e such that p e is the largest power of p that divides n!. This is the number of numbers from 1 to n divisible by p plus the number of numbers divisible by p 2 plus the number of numbers divisible by p 3 , etc. E (n,p) = floor [n/p] + floor [n/p 2] + floor [n/p 3] + ...
Using this definition with the product, P= (m+n)!/m!, we see that
E (m+n,p)-E (m,p) = floor [ (m+n)/p] - floor [m/p] + floor [ (m+n)/p 2] - floor [m/p 2] + floor [ (m+n)/p 3] - floor [m/p 3] + ...
which is at least as large as E (n,p) for any given m, n and p, because for any real numbers, a and b, floor [a+b] ≥ floor [a] + floor [b]
A note about products of negative numbers
The two proofs, above work for products of n positive integers, but the statement is also true when the product includes negative integers or zero. There are two cases that need to be considered: Case 1: the product of negative numbers. This is equal to plus or minus the product of the corresponding positive numbers, and so it is divisible by n!. Case 2: the product of numbers, some of which are negative, and some of which are not negative.
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http://2000clicks.com/MathHelp/BasicFactorialConsecutiveIntegerProducts.aspx
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Product of Consecutive Integers
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Product of Consecutive Integers
Product of Consecutive Integers
Contents of this page
Product of an Odd Number of Consecutive Integers
Product of an Even Number of Consecutive Integers
Ratio of Factorials, and Gosper's Approximation
The product of n consecutive integers is divisible by n!
A note about products of negative numbers
Proof that floor [a+b] ≥ floor [a] + floor [b]
Internet references
Related pages in this website
|
Case 1: the product of negative numbers. This is equal to plus or minus the product of the corresponding positive numbers, and so it is divisible by n!. Case 2: the product of numbers, some of which are negative, and some of which are not negative. In this case, one of the numbers must be zero, so the product is zero, and zero is divisible by all n! Proof that floor [a+b] ≥ floor [a] + floor [b]
It may seem obvious, but it's a little tricky, so I've included this little proof: a ≥ floor [a], and b ≥ floor [b], so
a+b ≥ floor [a]+floor [b]. floor is a nondecreasing function, which means if x ≥ y, then floor [x] ≥ floor [y], so we can take the "floor" of both sides of this inequality: floor [a+b] ≥ floor [floor [a]+floor [b]] = floor [a]+floor [b]
Internet references
Mathworld:
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http://2000clicks.com/MathHelp/BasicFactorialConsecutiveIntegerProducts.aspx
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Product of Consecutive Integers
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Product of Consecutive Integers
Product of Consecutive Integers
Contents of this page
Product of an Odd Number of Consecutive Integers
Product of an Even Number of Consecutive Integers
Ratio of Factorials, and Gosper's Approximation
The product of n consecutive integers is divisible by n!
A note about products of negative numbers
Proof that floor [a+b] ≥ floor [a] + floor [b]
Internet references
Related pages in this website
|
In this case, one of the numbers must be zero, so the product is zero, and zero is divisible by all n! Proof that floor [a+b] ≥ floor [a] + floor [b]
It may seem obvious, but it's a little tricky, so I've included this little proof: a ≥ floor [a], and b ≥ floor [b], so
a+b ≥ floor [a]+floor [b]. floor is a nondecreasing function, which means if x ≥ y, then floor [x] ≥ floor [y], so we can take the "floor" of both sides of this inequality: floor [a+b] ≥ floor [floor [a]+floor [b]] = floor [a]+floor [b]
Internet references
Mathworld: Factorial
Mathworld: Stirling's Approximation
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Puzzle: A Two-Player Game proves some interesting facts about complementary Beatty sequences using the floor function. The webmaster and author of this Math Help site is Graeme McRae .
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Quadrilaterals
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Quadrilaterals
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Math Help > Geometry > Quadrilaterals
Classification of quadrilaterals
Depiction of a hierarchy of quadrilaterals, in which the green arrows represent additional properties. For example, a parallelogram becomes a rhombus by making all four sides the same length. Similar charts can be found in Mathwords , Wikipedia, and Dr. Math. Figure
Definition, followed by additional facts (theorems)
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A closed plane figure consisting of four line segments. A "crossed" (or "complex") quadrilateral has one pair of intersecting sides. A "concave" quadrilateral, pictured here, has one interior angle greater than 180°. The other possibility, described next, is "convex". Bretschneider's Formula gives the area of a quadrilateral with sides of length a, b, c, d and opposite interior angles A and C. The sum of the interior angles of a non-crossed quadrilateral 360. In a crossed quadrilateral, the sum of the interior angles on one side of the crossing equals the sum of the interior angles on the other side of the crossing.
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Quadrilaterals
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Quadrilaterals
Quadrilaterals
Classification of quadrilaterals
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A "crossed" (or "complex") quadrilateral has one pair of intersecting sides. A "concave" quadrilateral, pictured here, has one interior angle greater than 180°. The other possibility, described next, is "convex". Bretschneider's Formula gives the area of a quadrilateral with sides of length a, b, c, d and opposite interior angles A and C. The sum of the interior angles of a non-crossed quadrilateral 360. In a crossed quadrilateral, the sum of the interior angles on one side of the crossing equals the sum of the interior angles on the other side of the crossing. Mathwords , Mathworld , Dr. Math , Wikipedia, Geometry Atlas, Math.com , Math is fun
"convex" means every line segment connecting interior points is entirely contained within the interior. Theorems: A case of Ramsey's Theorem tells us: Given any five points in a plane with no three collinear, four are the vertices of a convex quadrilateral.
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Quadrilaterals
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Quadrilaterals
Quadrilaterals
Classification of quadrilaterals
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Mathwords , Mathworld , Dr. Math , Wikipedia, Geometry Atlas, Math.com , Math is fun
"convex" means every line segment connecting interior points is entirely contained within the interior. Theorems: A case of Ramsey's Theorem tells us: Given any five points in a plane with no three collinear, four are the vertices of a convex quadrilateral. Mathwords , Dr. Math
A quadrilateral with two pairs of adjacent equal sides. ( In some text, a kite need not be convex; in others concave kites are termed a "dart" or "arrowhead".) Theorems:
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Quadrilaterals
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Quadrilaterals
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Mathwords , Dr. Math
A quadrilateral with two pairs of adjacent equal sides. ( In some text, a kite need not be convex; in others concave kites are termed a "dart" or "arrowhead".) Theorems: One set of opposite angles is equal, and that one diagonal perpendicularly bisects the other. The bisecting diagonal forms an axis of symmetry, dividing the kite into two congruent triangles. The bisected diagonal divides the kite into two isosceles triangles. A convex kite is tangential (inscriptable). A quadrilateral has bilateral symmetry iff it is either a kite or an isosceles trapezoid.
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One set of opposite angles is equal, and that one diagonal perpendicularly bisects the other. The bisecting diagonal forms an axis of symmetry, dividing the kite into two congruent triangles. The bisected diagonal divides the kite into two isosceles triangles. A convex kite is tangential (inscriptable). A quadrilateral has bilateral symmetry iff it is either a kite or an isosceles trapezoid. Mathwords , Mathworld, Dr. Math , Wikipedia, Geometry Atlas
A convex quadrilateral whose four vertices lie on a circumscribed circle. Theorems: In a cyclic quadrilateral, opposite angles are supplementary. The area of a cyclic quadrilateral is the maximum possible for any quadrilateral with the given side lengths.
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Mathwords , Mathworld, Dr. Math , Wikipedia, Geometry Atlas
A convex quadrilateral whose four vertices lie on a circumscribed circle. Theorems: In a cyclic quadrilateral, opposite angles are supplementary. The area of a cyclic quadrilateral is the maximum possible for any quadrilateral with the given side lengths. Brahmagupta's formula gives the area of a cyclic quadrilateral, using only the side lengths. Cyclic quadrilaterals that are also inscriptable are called "bicentric". Mathworld , Wikipedia , Geometry Atlas
A convex quadrilateral with one pair of parallel sides. Theorems: Two adjacent angles of a convex quadrilateral are supplementary iff it is a trapezoid.
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Brahmagupta's formula gives the area of a cyclic quadrilateral, using only the side lengths. Cyclic quadrilaterals that are also inscriptable are called "bicentric". Mathworld , Wikipedia , Geometry Atlas
A convex quadrilateral with one pair of parallel sides. Theorems: Two adjacent angles of a convex quadrilateral are supplementary iff it is a trapezoid. The diagonals cut each other in proportion of the lengths of the parallel sides. Mathwords , Mathworld, Dr. Math , Wikipedia, Geometry Atlas
A kite which is also cyclic. Theorems: A cyclic kite has a pair of opposite right angles.
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The diagonals cut each other in proportion of the lengths of the parallel sides. Mathwords , Mathworld, Dr. Math , Wikipedia, Geometry Atlas
A kite which is also cyclic. Theorems: A cyclic kite has a pair of opposite right angles. Being a kite, it has a pair of congruent angles. Being cyclic, these congruent angles must also be supplementary, so they are right angles. A trapezoid with two opposite sides parallel, the two other sides are of equal length. This implies that the two ends of each parallel side have equal angles, and that the diagonals are of equal length. A quadrilateral has bilateral symmetry iff it is either a kite or an isosceles trapezoid.
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Being a kite, it has a pair of congruent angles. Being cyclic, these congruent angles must also be supplementary, so they are right angles. A trapezoid with two opposite sides parallel, the two other sides are of equal length. This implies that the two ends of each parallel side have equal angles, and that the diagonals are of equal length. A quadrilateral has bilateral symmetry iff it is either a kite or an isosceles trapezoid. Mathwords , Mathworld , Wikipedia
A convex quadrilateral in which both pairs of opposite sides are parallel. This implies that opposite sides are of equal length, opposite angles are equal, and the diagonals bisect each other. Each diagonal bisects the parallelogram into two congruent triangles. Euclid showed if lines parallel to the sides are drawn through any point on a diagonal of a parallelogram, then the parallelograms not containing segments of that diagonal are equal in area (and conversely). Varignon's Theorem:
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Mathwords , Mathworld , Wikipedia
A convex quadrilateral in which both pairs of opposite sides are parallel. This implies that opposite sides are of equal length, opposite angles are equal, and the diagonals bisect each other. Each diagonal bisects the parallelogram into two congruent triangles. Euclid showed if lines parallel to the sides are drawn through any point on a diagonal of a parallelogram, then the parallelograms not containing segments of that diagonal are equal in area (and conversely). Varignon's Theorem: A parallelogram is formed by joining the midpoints of adjacent sides of a quadrilateral. The center of Varignon's Parallelogram is the centroid of the vertices of the quadrilateral. Wittenbauer's Theorem: A parallelogram is formed by dividing the sides of a quadrilateral into three equal parts, and connecting and extending adjacent points on either side of each vertex. The center of Wittenbauer's parallelogram is the quadrilateral's centroid.
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A parallelogram is formed by joining the midpoints of adjacent sides of a quadrilateral. The center of Varignon's Parallelogram is the centroid of the vertices of the quadrilateral. Wittenbauer's Theorem: A parallelogram is formed by dividing the sides of a quadrilateral into three equal parts, and connecting and extending adjacent points on either side of each vertex. The center of Wittenbauer's parallelogram is the quadrilateral's centroid. Mathwords , Mathworld , Wikipedia, Geometry Atlas
A convex quadrilateral with four right angles. This implies that opposite sides are parallel and of equal length, and the diagonals bisect each other and are equal in length. Mathworld , Wikipedia, Geometry Atlas
A convex quadrilateral with all four sides of equal length. This implies that opposite sides are parallel, opposite angles are equal, and the diagonals perpendicularly bisect each other. Its area is given by A=bh, where b is the base, and h is the height, or perpendicular distance between opposite sides.
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Mathwords , Mathworld , Wikipedia, Geometry Atlas
A convex quadrilateral with four right angles. This implies that opposite sides are parallel and of equal length, and the diagonals bisect each other and are equal in length. Mathworld , Wikipedia, Geometry Atlas
A convex quadrilateral with all four sides of equal length. This implies that opposite sides are parallel, opposite angles are equal, and the diagonals perpendicularly bisect each other. Its area is given by A=bh, where b is the base, and h is the height, or perpendicular distance between opposite sides. A rhombus is tangential (inscriptable). Mathwords , Mathworld, Wikipedia , Geometry Atlas
A convex quadrilateral with four sides of equal length, and four right angles. This implies that opposite sides are parallel, and that the diagonals perpendicularly bisect each other and are of equal length. Each diagonal bisects each pair of opposite angles. Mathwords , Mathworld, Wikipedia , Geometry Atlas
Tangential quadrilateral
Inscriptable quadrilateral
A convex quadrilateral in which a circle can be inscribed, tangent to all four sides.
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A rhombus is tangential (inscriptable). Mathwords , Mathworld, Wikipedia , Geometry Atlas
A convex quadrilateral with four sides of equal length, and four right angles. This implies that opposite sides are parallel, and that the diagonals perpendicularly bisect each other and are of equal length. Each diagonal bisects each pair of opposite angles. Mathwords , Mathworld, Wikipedia , Geometry Atlas
Tangential quadrilateral
Inscriptable quadrilateral
A convex quadrilateral in which a circle can be inscribed, tangent to all four sides. Theorems: The four angle bisectors meet (at the center of the inscribed circle) iff the figure is tangential. Pairs of opposites sides sum to the same number, which is the semiperimeter of the quadrilateral. Cyclic inscriptable quadrilaterals are also called "bicentric"
Wikipedia
A quadrilateral with just one pair of congruent opposite sides, and just one pair of congruent opposite angles is not necessarily a parallelogram. The reason, as explained in Dr. Math, can be seen by drawing the shorter diagonal in the figure to the left, which divides the figure into two non-congruent triangles which nonetheless have congruent side-side-angle (SSA).
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Theorems: The four angle bisectors meet (at the center of the inscribed circle) iff the figure is tangential. Pairs of opposites sides sum to the same number, which is the semiperimeter of the quadrilateral. Cyclic inscriptable quadrilaterals are also called "bicentric"
Wikipedia
A quadrilateral with just one pair of congruent opposite sides, and just one pair of congruent opposite angles is not necessarily a parallelogram. The reason, as explained in Dr. Math, can be seen by drawing the shorter diagonal in the figure to the left, which divides the figure into two non-congruent triangles which nonetheless have congruent side-side-angle (SSA). Dr. Math
Related pages in this website: Geometry Glossary defines geometrical terms. Bretschneider's Formula gives the area of a quadrilateral with sides of length a, b, c, d and opposite interior angles A and C.
Brahmagupta's formula gives the area of a cyclic quadrilateral, using only the side lengths. Internet references
Quadrilaterals are referenced by Mathwords , Mathworld , Dr.
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Dr. Math
Related pages in this website: Geometry Glossary defines geometrical terms. Bretschneider's Formula gives the area of a quadrilateral with sides of length a, b, c, d and opposite interior angles A and C.
Brahmagupta's formula gives the area of a cyclic quadrilateral, using only the side lengths. Internet references
Quadrilaterals are referenced by Mathwords , Mathworld , Dr. Math , Wikipedia, Geometry Atlas, Math.com , Math is fun.
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Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
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Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
Laws of Cosines and Sines
Triangle cases
AAS -- Law of Sines
SAS -- Law of Cosines
SSA -- Law of Sines
SSS -- Law of Cosines
Internet references
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Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
Math Help > Trigonometry > Law of Sines
In this page, we will consider a triangle ABC with sides a, b, and c. The triangle is labeled so that side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C.
Laws of Cosines and Sines
Law of Cosines
The Law of Cosines is most useful when you know two sides a, b, and the included angle, C: c² = a² + b² - 2ab cos (C)
or when you know all three sides: cos (C) = (a² + b² - c²) / (2ab)
Law of Sines
The Law of Sines is most useful when you know a side, a, and the angle, A, opposite it. Then for every other side you can find its opposite angle, and for every other angle, you can find its opposite side. b = a sin (B)/sin (A)
c = a sin (C)/sin (A)
sin (B) = (b/a) sin (A)
sin (C) = (c/a) sin (A)
Triangle cases
If you know any three facts about triangle ABC -- lengths of sides or measures of angles -- then you can use one of these laws to find the lengths of all sides and measures of all angles. These cases boil down to four cases: AAS -- Any two angles, and one side (Law of Sines)
SAS -- Side, included Angle, and Side (Law of Cosines)
SSA -- Two sides, and a non-included Angle (Law of Sines) — this might have two solutions! SSS -- All three sides (Law of Cosines)
Here is a breakdown of these four cases: AAS -- Law of Sines
If you know any two angles and any side, then you really know all three angles, so you have the "AAS" case -- Angle, Angle, Side. Let's say you know angles A, B, and C, and the side you know is side a. From the Law of Sines,
b/sin (B) = a/sin (A)
b = a sin (B) / sin (A)
example
a = 11, A=110°, B=20�, C=50�
From the Law of Sines,
b = a sin (B) / sin (A) = 11 * sin (20) / sin (110) = 4.00367
c = a sin (C) / sin (A) = 11 * sin (50) / sin (110) = 8.96728
SAS -- Law of Cosines
The Law of Cosines gives the length of the side opposite an angle if you know the lengths of the other two sides.
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Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
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Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
Laws of Cosines and Sines
Triangle cases
AAS -- Law of Sines
SAS -- Law of Cosines
SSA -- Law of Sines
SSS -- Law of Cosines
Internet references
Related pages in this website:
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These cases boil down to four cases: AAS -- Any two angles, and one side (Law of Sines)
SAS -- Side, included Angle, and Side (Law of Cosines)
SSA -- Two sides, and a non-included Angle (Law of Sines) — this might have two solutions! SSS -- All three sides (Law of Cosines)
Here is a breakdown of these four cases: AAS -- Law of Sines
If you know any two angles and any side, then you really know all three angles, so you have the "AAS" case -- Angle, Angle, Side. Let's say you know angles A, B, and C, and the side you know is side a. From the Law of Sines,
b/sin (B) = a/sin (A)
b = a sin (B) / sin (A)
example
a = 11, A=110°, B=20�, C=50�
From the Law of Sines,
b = a sin (B) / sin (A) = 11 * sin (20) / sin (110) = 4.00367
c = a sin (C) / sin (A) = 11 * sin (50) / sin (110) = 8.96728
SAS -- Law of Cosines
The Law of Cosines gives the length of the side opposite an angle if you know the lengths of the other two sides. This is the SAS case -- you know the Side, Angle, and Side. c² = a² + b² - 2ab cos (C)
example
Suppose you know a=58, b=120, and C=45�
The third side, c, is found using the law of cosines: c² = a² + b² - 2 a b cos (C)
58² + 120² - 2 * 58 * 120 * cos (45) = 7921.074, sqrt (7921.074) = 89.0004. Now that we know both side c and angle C we can use the Law of Sines to find the other two angles. This gives us:
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Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
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Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
Laws of Cosines and Sines
Triangle cases
AAS -- Law of Sines
SAS -- Law of Cosines
SSA -- Law of Sines
SSS -- Law of Cosines
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This is the SAS case -- you know the Side, Angle, and Side. c² = a² + b² - 2ab cos (C)
example
Suppose you know a=58, b=120, and C=45�
The third side, c, is found using the law of cosines: c² = a² + b² - 2 a b cos (C)
58² + 120² - 2 * 58 * 120 * cos (45) = 7921.074, sqrt (7921.074) = 89.0004. Now that we know both side c and angle C we can use the Law of Sines to find the other two angles. This gives us: sin (A) = (a/c) sin (C), and
sin (B) = (b/c) sin (C)
Now, watch out! It's possible that the largest angle of a triangle may be obtuse or acute, and you can't tell from the sine of an angle which it is, so here's a word to the wise: find the smaller angle first, knowing it's acute. So we'll use the Law of Sines to find the smaller angle A first, using
sin (A) = (a/c) sin (C). Plugging in the values we know,
sin (A) = (58 / 89.0004) sin (45) = 0.46081, so A = 27.43932
Now, finding angle B is easy:
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Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
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Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
Laws of Cosines and Sines
Triangle cases
AAS -- Law of Sines
SAS -- Law of Cosines
SSA -- Law of Sines
SSS -- Law of Cosines
Internet references
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sin (A) = (a/c) sin (C), and
sin (B) = (b/c) sin (C)
Now, watch out! It's possible that the largest angle of a triangle may be obtuse or acute, and you can't tell from the sine of an angle which it is, so here's a word to the wise: find the smaller angle first, knowing it's acute. So we'll use the Law of Sines to find the smaller angle A first, using
sin (A) = (a/c) sin (C). Plugging in the values we know,
sin (A) = (58 / 89.0004) sin (45) = 0.46081, so A = 27.43932
Now, finding angle B is easy: B = 180 - C - A, so B = 107.56068
SSA -- Law of Sines
The Law of Sines is a/ (sin A) = b/ (sin B) = c/ (sin C) = the diameter of the circumscribed circle. ( proof)
If you know the length of two sides and an angle other than the angle between those sides, then the Law of Sines can be used. This is the "SSA" case -- Side, Side, Angle. Assuming you know the lengths of sides a and b, and angle A,
a/sin (A) = b/sin (B)
sin (B) = (b/a) sin (A)
If a < b, and sin (B) = (b/a) sin (A) is between 0 and 1, then two different angles, B, can satisfy this equation: one is acute, the other is obtuse, and these two angles are supplementary.
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Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
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Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
Laws of Cosines and Sines
Triangle cases
AAS -- Law of Sines
SAS -- Law of Cosines
SSA -- Law of Sines
SSS -- Law of Cosines
Internet references
Related pages in this website:
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B = 180 - C - A, so B = 107.56068
SSA -- Law of Sines
The Law of Sines is a/ (sin A) = b/ (sin B) = c/ (sin C) = the diameter of the circumscribed circle. ( proof)
If you know the length of two sides and an angle other than the angle between those sides, then the Law of Sines can be used. This is the "SSA" case -- Side, Side, Angle. Assuming you know the lengths of sides a and b, and angle A,
a/sin (A) = b/sin (B)
sin (B) = (b/a) sin (A)
If a < b, and sin (B) = (b/a) sin (A) is between 0 and 1, then two different angles, B, can satisfy this equation: one is acute, the other is obtuse, and these two angles are supplementary. The two solutions of an SSA triangle
Here's an SSA triangle. Sides a (red) and b (green) are given, along with the non-included angle A. The conditions are right for two solutions — red shorter than green, and A small enough. The second diagram, to the right, shows the circumcircle of the triangle formed by the first solution, where the red line is CB. From the Law of Sines, we know the diameter of the circumcircle is a/sin (A) = b/sin (B), etc. So what about the second solution, where the red line is CD?
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Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
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Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
Laws of Cosines and Sines
Triangle cases
AAS -- Law of Sines
SAS -- Law of Cosines
SSA -- Law of Sines
SSS -- Law of Cosines
Internet references
Related pages in this website:
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The two solutions of an SSA triangle
Here's an SSA triangle. Sides a (red) and b (green) are given, along with the non-included angle A. The conditions are right for two solutions — red shorter than green, and A small enough. The second diagram, to the right, shows the circumcircle of the triangle formed by the first solution, where the red line is CB. From the Law of Sines, we know the diameter of the circumcircle is a/sin (A) = b/sin (B), etc. So what about the second solution, where the red line is CD? In both solutions, side a has the same length (it was given to us, after all), and angle A is fixed as well, so a/sin (A) has the same value for both solutions. This means the circumcircle of triangle ADC has the same diameter as the circumcircle of triangle ABC. Now, let's think about the converse: Suppose we draw two intersecting circles with the same diameter, and then we draw line AC connecting the circles' points of intersection. Then we draw any other line through point A that intersects both circles at points D and B. Now, by the Inscribed Angle Property, an inscribed angle, A, intercepts an arc whose measure is 2A. So the measure of arc CD in one circle equals the measure of arc CB in the other circle.
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Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
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Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
Laws of Cosines and Sines
Triangle cases
AAS -- Law of Sines
SAS -- Law of Cosines
SSA -- Law of Sines
SSS -- Law of Cosines
Internet references
Related pages in this website:
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In both solutions, side a has the same length (it was given to us, after all), and angle A is fixed as well, so a/sin (A) has the same value for both solutions. This means the circumcircle of triangle ADC has the same diameter as the circumcircle of triangle ABC. Now, let's think about the converse: Suppose we draw two intersecting circles with the same diameter, and then we draw line AC connecting the circles' points of intersection. Then we draw any other line through point A that intersects both circles at points D and B. Now, by the Inscribed Angle Property, an inscribed angle, A, intercepts an arc whose measure is 2A. So the measure of arc CD in one circle equals the measure of arc CB in the other circle. Since the circles have the same diameter, the chords CD and CB also have the same length. example
a=5, b=11, A=25�
We know side a and opposite angle A so we can use the Law of Sines to find angle B: sin (B) = (b/a) sin (A)
Plugging in the numbers gives us sin (B) = (11 / 5) sin (25) = 0.92976. Watch out! Since there are two different measures of angles B with the same sine, there may be two solutions to this problem.
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Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
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Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
Laws of Cosines and Sines
Triangle cases
AAS -- Law of Sines
SAS -- Law of Cosines
SSA -- Law of Sines
SSS -- Law of Cosines
Internet references
Related pages in this website:
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Since the circles have the same diameter, the chords CD and CB also have the same length. example
a=5, b=11, A=25�
We know side a and opposite angle A so we can use the Law of Sines to find angle B: sin (B) = (b/a) sin (A)
Plugging in the numbers gives us sin (B) = (11 / 5) sin (25) = 0.92976. Watch out! Since there are two different measures of angles B with the same sine, there may be two solutions to this problem. This can happen if side b is longer than side a, and sin (B), as calculated above, is between 0 and 1. In this case, since side b is indeed longer than side a, there are TWO possible angles for B: 68.39746�, and 111.60254�. For each possible measure of angle B, we use C = 180 - A - B, giving us 86.60254� or 43.39746�. Law of Sines OR Law of Cosines can be used to find the remaining side, c.
To find side c, for each possible angle C, we can use the law of cosines or the law of sines.
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Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
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Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
Laws of Cosines and Sines
Triangle cases
AAS -- Law of Sines
SAS -- Law of Cosines
SSA -- Law of Sines
SSS -- Law of Cosines
Internet references
Related pages in this website:
|
This can happen if side b is longer than side a, and sin (B), as calculated above, is between 0 and 1. In this case, since side b is indeed longer than side a, there are TWO possible angles for B: 68.39746�, and 111.60254�. For each possible measure of angle B, we use C = 180 - A - B, giving us 86.60254� or 43.39746�. Law of Sines OR Law of Cosines can be used to find the remaining side, c.
To find side c, for each possible angle C, we can use the law of cosines or the law of sines. Using the Law of Cosines, c² = a² + b² - 2 a b cos (C)
Using the Law of Sines, c = a sin (C)/sin (A)
Using either method, the solutions are c = 11.81021, or c = 8.12856. SSS -- Law of Cosines
If you know the lengths of all three sides, you can solve for any angle: A = acos ( (b²+c²-a²)/ (2bc))
B = acos ( (c²+a²-b²)/ (2ca))
C = acos ( (a²+b²-c²)/ (2ab))
example
Suppose a=56, b=97, c=112. A = acos ( (b²+c²-a²)/ (2bc)) = 29.99999986�
B = acos ( (c²+a²-b²)/ (2ca)) = 60.00527405�
C = acos ( (a²+b²-c²)/ (2ab)) = 89.99472609�
This is an amusing example, because it's a triangle that's almost a right triangle: a²+b²=12545, and c²=12544.
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http://2000clicks.com/MathHelp/GeometryLawOfSines.aspx
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Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
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Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
Laws of Cosines and Sines
Triangle cases
AAS -- Law of Sines
SAS -- Law of Cosines
SSA -- Law of Sines
SSS -- Law of Cosines
Internet references
Related pages in this website:
|
Using the Law of Cosines, c² = a² + b² - 2 a b cos (C)
Using the Law of Sines, c = a sin (C)/sin (A)
Using either method, the solutions are c = 11.81021, or c = 8.12856. SSS -- Law of Cosines
If you know the lengths of all three sides, you can solve for any angle: A = acos ( (b²+c²-a²)/ (2bc))
B = acos ( (c²+a²-b²)/ (2ca))
C = acos ( (a²+b²-c²)/ (2ab))
example
Suppose a=56, b=97, c=112. A = acos ( (b²+c²-a²)/ (2bc)) = 29.99999986�
B = acos ( (c²+a²-b²)/ (2ca)) = 60.00527405�
C = acos ( (a²+b²-c²)/ (2ab)) = 89.99472609�
This is an amusing example, because it's a triangle that's almost a right triangle: a²+b²=12545, and c²=12544. In addition, this triangle has almost a 60� angle: c²+a²-b²=6271, and 2ca=6272. To find more examples of near 30-60-90 triangles, look for the continued fraction convergents to sqrt (3). a is an element of A002530,
b is an element of A002531, and
c = 2a
Internet references
Mathworld: AAS , SAS , ASS, and SSS theorems.
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http://2000clicks.com/MathHelp/GeometryLawOfSines.aspx
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Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
|
Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
Laws of Cosines and Sines
Triangle cases
AAS -- Law of Sines
SAS -- Law of Cosines
SSA -- Law of Sines
SSS -- Law of Cosines
Internet references
Related pages in this website:
|
In addition, this triangle has almost a 60� angle: c²+a²-b²=6271, and 2ca=6272. To find more examples of near 30-60-90 triangles, look for the continued fraction convergents to sqrt (3). a is an element of A002530,
b is an element of A002531, and
c = 2a
Internet references
Mathworld: AAS , SAS , ASS, and SSS theorems. OEIS: A002530, A002531, denominators and numerators of convergents to sqrt (3). Related pages in this website: Law of Sines Proof
Law of Cosines : c² = a² + b² − 2ab cos C
Inscribed Angle Property -- that all angles that are inscribed in a circle that are subtended by a given chord have equal measure, and that measure is half the central angle subtended by the same chord.
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http://2000clicks.com/MathHelp/GeometryLawOfSines.aspx
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Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
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Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
Solving a triangle (AAS, SAS, SSA, SSS) using Laws of Sines and Cosines
Laws of Cosines and Sines
Triangle cases
AAS -- Law of Sines
SAS -- Law of Cosines
SSA -- Law of Sines
SSS -- Law of Cosines
Internet references
Related pages in this website:
|
OEIS: A002530, A002531, denominators and numerators of convergents to sqrt (3). Related pages in this website: Law of Sines Proof
Law of Cosines : c² = a² + b² − 2ab cos C
Inscribed Angle Property -- that all angles that are inscribed in a circle that are subtended by a given chord have equal measure, and that measure is half the central angle subtended by the same chord. The webmaster and author of this Math Help site is Graeme McRae .
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http://2000clicks.com/MathHelp/GeometryTriangleCenterInscribedCircle2.aspx
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Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
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Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
Concurrence of the angle bisectors
Barycentric coordinates of the incenter
Angle Bisectors
Inscribed Circle
Excribed Circles
Additional Angle Bisector Properties
Internet references
Related pages in this website
|
Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
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Math Help > Geometry > Polygons and Triangles > Triangle Centers > Inscribed Circle and Angle Bisectors
The incenter of a triangle is the point of intersection of the triangle's three angle bisectors. Incenter, concurrency of the three angle bisectors
Concurrence of the angle bisectors
The three angle bisectors have to meet in a single point because...
If a circle is inscribed in an angle, the angle bisector passes through the center of the circle. The two sides of the angle each form a "point of tangency" where they intersect the circle (not shown on the diagram). At each point of tangency, there is a radius that meets the side of the angle at a right angle. You can there are two congruent triangles formed by the vertex of the angle, a point of tangency, and the intersection of the two radii, which, of course has to be at the center of the circle. There are several interesting relationships in a triangle between the inscribed circle, the angle bisectors, and the three "exscribed" circles. Barycentric coordinates of the incenter
As a review of the barycentric coordinates of point P in triangle ABC, I'll remind you they are the three weights you need to give points A, B, and C so that P is the centroid (weighted average) of the three vertices. So, for example, A= (1,0,0), B= (0,1,0), and C= (0,0,1).
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http://2000clicks.com/MathHelp/GeometryTriangleCenterInscribedCircle2.aspx
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Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
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Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
Concurrence of the angle bisectors
Barycentric coordinates of the incenter
Angle Bisectors
Inscribed Circle
Excribed Circles
Additional Angle Bisector Properties
Internet references
Related pages in this website
|
At each point of tangency, there is a radius that meets the side of the angle at a right angle. You can there are two congruent triangles formed by the vertex of the angle, a point of tangency, and the intersection of the two radii, which, of course has to be at the center of the circle. There are several interesting relationships in a triangle between the inscribed circle, the angle bisectors, and the three "exscribed" circles. Barycentric coordinates of the incenter
As a review of the barycentric coordinates of point P in triangle ABC, I'll remind you they are the three weights you need to give points A, B, and C so that P is the centroid (weighted average) of the three vertices. So, for example, A= (1,0,0), B= (0,1,0), and C= (0,0,1). Another way to view barycentric coordinates is as "proportional altitudes". Let me explain. A point, P, can be identified by its distance from each of the three sides as a proportion of the altitude to that side. So, if h A is the altitude of point A from its opposite side, BC, and h B and h C are the other altitudes, then barycentric coordinates of a point Q = (x,y,z) indicate that the distances of Q from the three sides are x�h A, y�h B, z�h C, respectively. Now, the incenter is equidistant from each of the sides, distant from each side by the length of the radius of the incircle.
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http://2000clicks.com/MathHelp/GeometryTriangleCenterInscribedCircle2.aspx
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Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
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Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
Concurrence of the angle bisectors
Barycentric coordinates of the incenter
Angle Bisectors
Inscribed Circle
Excribed Circles
Additional Angle Bisector Properties
Internet references
Related pages in this website
|
Another way to view barycentric coordinates is as "proportional altitudes". Let me explain. A point, P, can be identified by its distance from each of the three sides as a proportion of the altitude to that side. So, if h A is the altitude of point A from its opposite side, BC, and h B and h C are the other altitudes, then barycentric coordinates of a point Q = (x,y,z) indicate that the distances of Q from the three sides are x�h A, y�h B, z�h C, respectively. Now, the incenter is equidistant from each of the sides, distant from each side by the length of the radius of the incircle. So, viewing the barycentric coordinates as proportional altitudes, and letting the incenter I = (x, y, z), we see that in order to put point I equidistant from the three sides, we need x=1/h A, y=1/h B, and z=1/h C, so the barycentric coordinates of I are
(1/h A, 1/h B, 1/h C)
The scale of each of the points is arbitrary, because these barycentric coordinates are not normalized. So, for example, the barycentric coordinates (1,2,3) and (2,4,6) represent the same point. Now, observe that the altitudes h A, h B, h C are inversely proportional to their bases, because the product of base�altitude is constant, the area of the triangle. So the barycentric coordinates of the incenter can be more simply represented,
(a, b, c)
You can calculate the vector coordinates of the incenter from its barycentric coordinates by multiplying each vertex (as a vector) by the corresponding barycentric coordinate, adding the results, and then dividing by the sum of the barycentric coordinates. If the vertices of the triangle are given on a two-dimensional plane as A (a,b), B (c,d), and C (e,f) then the incenter (h,k) is given by
h = (a sqrt ( (c-e) 2 + (d-f) 2) + c sqrt ( (e-a) 2 + (f-b) 2) + e sqrt ( (a-c) 2 + (b-d) 2) ) /
(sqrt ( (c-e) 2 + (d-f) 2) + sqrt ( (e-a) 2 + (f-b) 2) + sqrt ( (a-c) 2 + (b-d) 2) )
k = (b sqrt ( (c-e) 2 + (d-f) 2) + d sqrt ( (e-a) 2 + (f-b) 2) + f sqrt ( (a-c) 2 + (b-d) 2) ) /
(sqrt ( (c-e) 2 + (d-f) 2) + sqrt ( (e-a) 2 + (f-b) 2) + sqrt ( (a-c) 2 + (b-d) 2) )
Do you see how we derived the two equations, above, from the barycentric coordinates of the incenter?
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http://2000clicks.com/MathHelp/GeometryTriangleCenterInscribedCircle2.aspx
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Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
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Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
Concurrence of the angle bisectors
Barycentric coordinates of the incenter
Angle Bisectors
Inscribed Circle
Excribed Circles
Additional Angle Bisector Properties
Internet references
Related pages in this website
|
So, viewing the barycentric coordinates as proportional altitudes, and letting the incenter I = (x, y, z), we see that in order to put point I equidistant from the three sides, we need x=1/h A, y=1/h B, and z=1/h C, so the barycentric coordinates of I are
(1/h A, 1/h B, 1/h C)
The scale of each of the points is arbitrary, because these barycentric coordinates are not normalized. So, for example, the barycentric coordinates (1,2,3) and (2,4,6) represent the same point. Now, observe that the altitudes h A, h B, h C are inversely proportional to their bases, because the product of base�altitude is constant, the area of the triangle. So the barycentric coordinates of the incenter can be more simply represented,
(a, b, c)
You can calculate the vector coordinates of the incenter from its barycentric coordinates by multiplying each vertex (as a vector) by the corresponding barycentric coordinate, adding the results, and then dividing by the sum of the barycentric coordinates. If the vertices of the triangle are given on a two-dimensional plane as A (a,b), B (c,d), and C (e,f) then the incenter (h,k) is given by
h = (a sqrt ( (c-e) 2 + (d-f) 2) + c sqrt ( (e-a) 2 + (f-b) 2) + e sqrt ( (a-c) 2 + (b-d) 2) ) /
(sqrt ( (c-e) 2 + (d-f) 2) + sqrt ( (e-a) 2 + (f-b) 2) + sqrt ( (a-c) 2 + (b-d) 2) )
k = (b sqrt ( (c-e) 2 + (d-f) 2) + d sqrt ( (e-a) 2 + (f-b) 2) + f sqrt ( (a-c) 2 + (b-d) 2) ) /
(sqrt ( (c-e) 2 + (d-f) 2) + sqrt ( (e-a) 2 + (f-b) 2) + sqrt ( (a-c) 2 + (b-d) 2) )
Do you see how we derived the two equations, above, from the barycentric coordinates of the incenter? If not, consider this: The calculation, above, is easier than it looks to carry out, because of the common subexpressions for the lengths of the sides. If we let the three side lengths be
A = sqrt ( (c-e) 2 + (d-f) 2 ),
B = sqrt ( (e-a) 2 + (f-b) 2 ), and
C = sqrt ( (a-c) 2 + (b-d) 2 ),
then the values of h and k can be more simply represented as
h = (aA + cB + eC ) / (A+B+C), and
k = (bA + dB + fC ) / (A+B+C)
Angle Bisectors
Consider triangle ABC, pictured on the left side of this page. Now, if we inscribe a circle in any of its angles, say, angle ACB, then the center, O, of the circle bisects the angle. This is true because of symmetry:
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http://2000clicks.com/MathHelp/GeometryTriangleCenterInscribedCircle2.aspx
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Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
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Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
Concurrence of the angle bisectors
Barycentric coordinates of the incenter
Angle Bisectors
Inscribed Circle
Excribed Circles
Additional Angle Bisector Properties
Internet references
Related pages in this website
|
If not, consider this: The calculation, above, is easier than it looks to carry out, because of the common subexpressions for the lengths of the sides. If we let the three side lengths be
A = sqrt ( (c-e) 2 + (d-f) 2 ),
B = sqrt ( (e-a) 2 + (f-b) 2 ), and
C = sqrt ( (a-c) 2 + (b-d) 2 ),
then the values of h and k can be more simply represented as
h = (aA + cB + eC ) / (A+B+C), and
k = (bA + dB + fC ) / (A+B+C)
Angle Bisectors
Consider triangle ABC, pictured on the left side of this page. Now, if we inscribe a circle in any of its angles, say, angle ACB, then the center, O, of the circle bisects the angle. This is true because of symmetry: The quadrilateral formed by C, O, and the two points where the circle is tangent to the angle is symmetrical about line CO, which means angle ACO is equal to angle BCO. So the center of any circle inscribed in an angle defines the angle bisector of that angle. The reverse is also true. That is, if O is any point in the interior of an angle such that ray CO bisects angle ACB, then the circle with its center at O that is tangent to AC is also tangent to BC. Let's take this idea a step further.
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http://2000clicks.com/MathHelp/GeometryTriangleCenterInscribedCircle2.aspx
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Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
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Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
Concurrence of the angle bisectors
Barycentric coordinates of the incenter
Angle Bisectors
Inscribed Circle
Excribed Circles
Additional Angle Bisector Properties
Internet references
Related pages in this website
|
The quadrilateral formed by C, O, and the two points where the circle is tangent to the angle is symmetrical about line CO, which means angle ACO is equal to angle BCO. So the center of any circle inscribed in an angle defines the angle bisector of that angle. The reverse is also true. That is, if O is any point in the interior of an angle such that ray CO bisects angle ACB, then the circle with its center at O that is tangent to AC is also tangent to BC. Let's take this idea a step further. If we draw two angle bisectors, one bisecting angle C, and the other bisecting angle B, then they will intersect at some point inside the triangle, which we will label point "I". Since point I is on the bisector of angle C, the circle centered at I and tangent to AC is also tangent to BC. Also, since point I is on the bisector of angle B, that same circle, tangent to AC, will also be tangent to AB. So, you see, this circle is tangent to all three sides of the triangle. Inscribed Circle
From this we see that the intersection of any two angle bisectors is the center if the inscribed circle.
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http://2000clicks.com/MathHelp/GeometryTriangleCenterInscribedCircle2.aspx
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Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
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Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
Concurrence of the angle bisectors
Barycentric coordinates of the incenter
Angle Bisectors
Inscribed Circle
Excribed Circles
Additional Angle Bisector Properties
Internet references
Related pages in this website
|
If we draw two angle bisectors, one bisecting angle C, and the other bisecting angle B, then they will intersect at some point inside the triangle, which we will label point "I". Since point I is on the bisector of angle C, the circle centered at I and tangent to AC is also tangent to BC. Also, since point I is on the bisector of angle B, that same circle, tangent to AC, will also be tangent to AB. So, you see, this circle is tangent to all three sides of the triangle. Inscribed Circle
From this we see that the intersection of any two angle bisectors is the center if the inscribed circle. It follows that all three internal angle bisectors intersect at one point, which is the center of the inscribed circle, or "incircle". The perpendicular distance from point I to any of the sides is the radius, r, of the incircle. The area, K, of triangle ABC is the sum of the areas of AIB, BIC, and CIA. If we label the lengths of the sides of triangle ABC in traditional form, with side a opposite vertex A, b opposite B, and c opposite C, then the areas of these three small triangles are cr/2, ar/2, and br/2 because the height of each small triangle is the radius of the incircle. The sum of these areas is
K = (a+b+c) (r)/2, or
K = sr.
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http://2000clicks.com/MathHelp/GeometryTriangleCenterInscribedCircle2.aspx
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Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
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Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
Concurrence of the angle bisectors
Barycentric coordinates of the incenter
Angle Bisectors
Inscribed Circle
Excribed Circles
Additional Angle Bisector Properties
Internet references
Related pages in this website
|
It follows that all three internal angle bisectors intersect at one point, which is the center of the inscribed circle, or "incircle". The perpendicular distance from point I to any of the sides is the radius, r, of the incircle. The area, K, of triangle ABC is the sum of the areas of AIB, BIC, and CIA. If we label the lengths of the sides of triangle ABC in traditional form, with side a opposite vertex A, b opposite B, and c opposite C, then the areas of these three small triangles are cr/2, ar/2, and br/2 because the height of each small triangle is the radius of the incircle. The sum of these areas is
K = (a+b+c) (r)/2, or
K = sr. Also,
r = K/s
where s = (a+b+c)/2 is the semiperimeter of triangle ABC. By extending two of the sides of the triangle, AC, and AB, we can bisect the exterior angles of the triangle as well. Here we have drawn two exterior bisectors which intersect at point E a . The same logic that we used for the incircle can be used again to show that a circle can be drawn at E a that is tangent to all three sides, suitably extended, of the triangle. A circle centered at E a and tangent to AC is also tangent to CB, because E a is on the bisector of the angle formed by those two lines.
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http://2000clicks.com/MathHelp/GeometryTriangleCenterInscribedCircle2.aspx
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Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
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Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
Concurrence of the angle bisectors
Barycentric coordinates of the incenter
Angle Bisectors
Inscribed Circle
Excribed Circles
Additional Angle Bisector Properties
Internet references
Related pages in this website
|
Also,
r = K/s
where s = (a+b+c)/2 is the semiperimeter of triangle ABC. By extending two of the sides of the triangle, AC, and AB, we can bisect the exterior angles of the triangle as well. Here we have drawn two exterior bisectors which intersect at point E a . The same logic that we used for the incircle can be used again to show that a circle can be drawn at E a that is tangent to all three sides, suitably extended, of the triangle. A circle centered at E a and tangent to AC is also tangent to CB, because E a is on the bisector of the angle formed by those two lines. Similarly, the same circle, which I remind you is tangent to CB is also tangent to AB because it lies on the external bisector of angle B. Finally, since this circle is tangent to both AC and AB, the internal bisector of angle A also passes through point E a . This circle is called an "excribed" circle, or excircle. Excribed Circles
Just as the three "in-triangles" AIB, BIC, and CIA add up to triangle ABC, three "ex-triangles" AE a B, BE a C, and CE a A add up, in a way, to the same triangle ABC. That is, if you subtract the area of BE a C from the sum of the areas AE a B and CE a A, the result is the area of ABC. K = (-a+b+c) (R a )/2, or
K = (s-a) (R a ).
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http://2000clicks.com/MathHelp/GeometryTriangleCenterInscribedCircle2.aspx
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Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
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Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
Concurrence of the angle bisectors
Barycentric coordinates of the incenter
Angle Bisectors
Inscribed Circle
Excribed Circles
Additional Angle Bisector Properties
Internet references
Related pages in this website
|
Similarly, the same circle, which I remind you is tangent to CB is also tangent to AB because it lies on the external bisector of angle B. Finally, since this circle is tangent to both AC and AB, the internal bisector of angle A also passes through point E a . This circle is called an "excribed" circle, or excircle. Excribed Circles
Just as the three "in-triangles" AIB, BIC, and CIA add up to triangle ABC, three "ex-triangles" AE a B, BE a C, and CE a A add up, in a way, to the same triangle ABC. That is, if you subtract the area of BE a C from the sum of the areas AE a B and CE a A, the result is the area of ABC. K = (-a+b+c) (R a )/2, or
K = (s-a) (R a ). Also,
R a = K/ (s-a)
where s is the semiperimeter of triangle ABC, and Ra is the radius of the excircle with center E a . Why stop there? Both external bisectors of angle C are on the same line, as are those of angles B and A. In this diagram, these bisectors were extended to show two other excribed circles. As you can see, a triangle has three internal angle bisectors and three external angle bisectors. These six lines intersect in exactly seven points:
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Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
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Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
Concurrence of the angle bisectors
Barycentric coordinates of the incenter
Angle Bisectors
Inscribed Circle
Excribed Circles
Additional Angle Bisector Properties
Internet references
Related pages in this website
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Also,
R a = K/ (s-a)
where s is the semiperimeter of triangle ABC, and Ra is the radius of the excircle with center E a . Why stop there? Both external bisectors of angle C are on the same line, as are those of angles B and A. In this diagram, these bisectors were extended to show two other excribed circles. As you can see, a triangle has three internal angle bisectors and three external angle bisectors. These six lines intersect in exactly seven points: the three vertices of the triangle (pairwise intersections), the centers of the four in- and excircles (triple intersections). The internal angle bisector of a given vertex is perpendicular to the external angle bisector. This makes each of the internal angle bisectors an altitude of triangle E a E b E c formed by the three excenters. The point I then, which is the incenter of triangle ABC, is also the orthocenter of triangle E a E b E c.
The radii of the four circles shown are related by the equation
1/r = 1/R a +1/R b +1/R c because R a = K/ (s-a), etc. so
1/R a = (s-a)/K;
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Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
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Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
Concurrence of the angle bisectors
Barycentric coordinates of the incenter
Angle Bisectors
Inscribed Circle
Excribed Circles
Additional Angle Bisector Properties
Internet references
Related pages in this website
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the three vertices of the triangle (pairwise intersections), the centers of the four in- and excircles (triple intersections). The internal angle bisector of a given vertex is perpendicular to the external angle bisector. This makes each of the internal angle bisectors an altitude of triangle E a E b E c formed by the three excenters. The point I then, which is the incenter of triangle ABC, is also the orthocenter of triangle E a E b E c.
The radii of the four circles shown are related by the equation
1/r = 1/R a +1/R b +1/R c because R a = K/ (s-a), etc. so
1/R a = (s-a)/K; 1/R b = (s-b)/K; 1/R c = (s-c)/K, so
1/R a +1/R b +1/R c = (3s-a-b-c)/K = s/K = 1/r
The radii of the four circles shown are related to the area of triangle ABC by the formula K=sqrt (r R a R b R c ), because from Heron's Formula
K 2 = s (s-a) (s-b) (s-c). So,
sqrt (r R a R b R c) =
sqrt (K/s K/ (s-a) K/ (s-b) K/ (s-c)) =
sqrt (K 4 / (s (s-a) (s-b) (s-c)) ) =
sqrt (K 4 / K 2) = K.
If we let R stand for the radius of the circle that circumscibes triangle ABC (the larger gold circle in the diagram, below), we get one more interesting theorem, R a +R b +R c -r=4R. This is because 4R=abc/K (proof) along with the following rather tedious algebra: R a +R b +R c -r = K/ (s-a) + K/ (s-b) + K/ (s-c) - K/s
now, multiplying each fraction by 1 = s (s-a) (s-b) (s-c)/K 2,
R a +R b +R c -r = s (s-b) (s-c)/K + s (s-a) (s-c)/K + s (s-a) (s-b)/K - (s-a) (s-b) (s-c)/K
= (s 3 -bs 2 -cs 2 +bcs+s 3 -as 2 -cs 2 +acs+s 3 -as 2 -bs 2 +abs-s 3 +as 2 +bs 2 +cs 2 -abs-acs-bcs+abc)/K
= (2s 3 - as 2 - bs 2 - cs 2 + abc)/K
= (2s 3 - (a+b+c)s 2 + abc)/K
= abc/K
= 4R
The three excircles, E a, E b, and E c are externally tangent to the Nine Point Circle, the smaller gold circle shown here. The Nine Point Circle is so-called because it passes through the three "feet" of the altitudes (or their extensions), the midpoints of the three sides, and the midpoints of the segments connecting each of the vertices to the orthocenter (where the three altitudes meet).
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Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
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Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
Incenter: concurrency of the angle bisectors, Angle Bisector Theorem
Concurrence of the angle bisectors
Barycentric coordinates of the incenter
Angle Bisectors
Inscribed Circle
Excribed Circles
Additional Angle Bisector Properties
Internet references
Related pages in this website
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1/R b = (s-b)/K; 1/R c = (s-c)/K, so
1/R a +1/R b +1/R c = (3s-a-b-c)/K = s/K = 1/r
The radii of the four circles shown are related to the area of triangle ABC by the formula K=sqrt (r R a R b R c ), because from Heron's Formula
K 2 = s (s-a) (s-b) (s-c). So,
sqrt (r R a R b R c) =
sqrt (K/s K/ (s-a) K/ (s-b) K/ (s-c)) =
sqrt (K 4 / (s (s-a) (s-b) (s-c)) ) =
sqrt (K 4 / K 2) = K.
If we let R stand for the radius of the circle that circumscibes triangle ABC (the larger gold circle in the diagram, below), we get one more interesting theorem, R a +R b +R c -r=4R. This is because 4R=abc/K (proof) along with the following rather tedious algebra: R a +R b +R c -r = K/ (s-a) + K/ (s-b) + K/ (s-c) - K/s
now, multiplying each fraction by 1 = s (s-a) (s-b) (s-c)/K 2,
R a +R b +R c -r = s (s-b) (s-c)/K + s (s-a) (s-c)/K + s (s-a) (s-b)/K - (s-a) (s-b) (s-c)/K
= (s 3 -bs 2 -cs 2 +bcs+s 3 -as 2 -cs 2 +acs+s 3 -as 2 -bs 2 +abs-s 3 +as 2 +bs 2 +cs 2 -abs-acs-bcs+abc)/K
= (2s 3 - as 2 - bs 2 - cs 2 + abc)/K
= (2s 3 - (a+b+c)s 2 + abc)/K
= abc/K
= 4R
The three excircles, E a, E b, and E c are externally tangent to the Nine Point Circle, the smaller gold circle shown here. The Nine Point Circle is so-called because it passes through the three "feet" of the altitudes (or their extensions), the midpoints of the three sides, and the midpoints of the segments connecting each of the vertices to the orthocenter (where the three altitudes meet). In addition to being externally tangent to the three excircles, the Nine Point Circle is internally tangent to the incircle at a point called the Feuerbach point. This result is known as
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Number Theory Definitions and Principles
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Number Theory Definitions and Principles
Number Theory Definitions and Principles
Number Theory Definitions and Principles
Modulo Arithmetic
Division
Some properties of the Greatest Common Divisor (GCD)
Definition of "divides"
Definition of GCD, "Greatest Common Divisor"
Well-Ordering Principle
Division Algorithm Property
Linear Combination Property
Euclidean Algorithm Property, a.k.a. B�zout's Identity.
Euclid's principle:
Exercise: Proof
Other Properties of GCD
Internet references
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Number Theory Definitions and Principles
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Math Help > Number Theory > Definitions in Number Theory
Number Theory Definitions and Principles
This page has some basic definitions and proofs in number theory. Click one of the links below to jump to a section of this page: Modulo Arithmetic
Some properties of the Greatest Common Divisor (GCD)
Definition of "Divides": a|b if ak=b for some integer k
Definition of GCD: GCD (a,b)>=0, GCD (a,b)|a, GCD (a,b)|b, and if d|a and d|b then d|GCD (a,b). Well-Ordering Principle: says that a set of positive integers has a smallest element
Division Algorithm: If a and m are any integers with m not zero,
then there are unique integers q and r such that a = qm+r with 0 < r < |m|. Linear Combination: If a|b and a|c then a| (bx + cy) for any integers x and y.
Euclidean Algorithm Property:
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Number Theory Definitions and Principles
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Number Theory Definitions and Principles
Number Theory Definitions and Principles
Number Theory Definitions and Principles
Modulo Arithmetic
Division
Some properties of the Greatest Common Divisor (GCD)
Definition of "divides"
Definition of GCD, "Greatest Common Divisor"
Well-Ordering Principle
Division Algorithm Property
Linear Combination Property
Euclidean Algorithm Property, a.k.a. B�zout's Identity.
Euclid's principle:
Exercise: Proof
Other Properties of GCD
Internet references
|
Well-Ordering Principle: says that a set of positive integers has a smallest element
Division Algorithm: If a and m are any integers with m not zero,
then there are unique integers q and r such that a = qm+r with 0 < r < |m|. Linear Combination: If a|b and a|c then a| (bx + cy) for any integers x and y.
Euclidean Algorithm Property: For any integers a and b, there exist integers x and y such that GCD (a,b)=ax+by. Euclid's Principle: If a prime p|ab then p|a or p|b, a and b integers
Application of these principles: an exercise
Related pages in this website
Modulo Arithmetic
The MOD operator is defined such that
0 <= x MOD y < y and
x = q*y + (x MOD y)
In other words x MOD y is the remainder after dividing x by y.
The divides relation is defined such that x divides y <=> y = x * q for some integer q
Its properties include the following
x divides 0
x divides x
x divides y ==> x divides -y
x divides y and x divides z ==> x divides y + z
x divides y - (y MOD z)
The congruence operator == is defined such that
x == y (mod z) <=> z divides (x - y)
or equivalently
x == y (mod z) <=> x MOD z = y MOD z
z == 0 (mod z)
x == y (mod z) ==> y == x (mod z)
x == y (mod z) and y == t (mod z) ==> x == t (mod z)
x1 == x2 (mod z) and y1 == y2 (mod z) ==>
x1 + y1 == x2 + y2 and
x1 - y1 == x2 - y2 and
x1 * y2 == x2 * y2
Thus you can add, subtract, and multiply congruences. Division
Division Theorem:
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Number Theory Definitions and Principles
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Number Theory Definitions and Principles
Number Theory Definitions and Principles
Number Theory Definitions and Principles
Modulo Arithmetic
Division
Some properties of the Greatest Common Divisor (GCD)
Definition of "divides"
Definition of GCD, "Greatest Common Divisor"
Well-Ordering Principle
Division Algorithm Property
Linear Combination Property
Euclidean Algorithm Property, a.k.a. B�zout's Identity.
Euclid's principle:
Exercise: Proof
Other Properties of GCD
Internet references
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For any integers a and b, there exist integers x and y such that GCD (a,b)=ax+by. Euclid's Principle: If a prime p|ab then p|a or p|b, a and b integers
Application of these principles: an exercise
Related pages in this website
Modulo Arithmetic
The MOD operator is defined such that
0 <= x MOD y < y and
x = q*y + (x MOD y)
In other words x MOD y is the remainder after dividing x by y.
The divides relation is defined such that x divides y <=> y = x * q for some integer q
Its properties include the following
x divides 0
x divides x
x divides y ==> x divides -y
x divides y and x divides z ==> x divides y + z
x divides y - (y MOD z)
The congruence operator == is defined such that
x == y (mod z) <=> z divides (x - y)
or equivalently
x == y (mod z) <=> x MOD z = y MOD z
z == 0 (mod z)
x == y (mod z) ==> y == x (mod z)
x == y (mod z) and y == t (mod z) ==> x == t (mod z)
x1 == x2 (mod z) and y1 == y2 (mod z) ==>
x1 + y1 == x2 + y2 and
x1 - y1 == x2 - y2 and
x1 * y2 == x2 * y2
Thus you can add, subtract, and multiply congruences. Division
Division Theorem: If a and b are integers, b ≠ 0, a unique integer q exists such that
a = bq+r, where
0 ≤ r < |b|
Here, r is called the least non-negative remainder or principle remainder of a (mod b). r=0 iff b|a. Some properties of the Greatest Common Divisor (GCD)
See the definition of GCD, below, for its defining properties. Some other properties include: GCD (m,n) = max {d :
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Number Theory Definitions and Principles
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Number Theory Definitions and Principles
Number Theory Definitions and Principles
Number Theory Definitions and Principles
Modulo Arithmetic
Division
Some properties of the Greatest Common Divisor (GCD)
Definition of "divides"
Definition of GCD, "Greatest Common Divisor"
Well-Ordering Principle
Division Algorithm Property
Linear Combination Property
Euclidean Algorithm Property, a.k.a. B�zout's Identity.
Euclid's principle:
Exercise: Proof
Other Properties of GCD
Internet references
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If a and b are integers, b ≠ 0, a unique integer q exists such that
a = bq+r, where
0 ≤ r < |b|
Here, r is called the least non-negative remainder or principle remainder of a (mod b). r=0 iff b|a. Some properties of the Greatest Common Divisor (GCD)
See the definition of GCD, below, for its defining properties. Some other properties include: GCD (m,n) = max {d : d divides m and d divides n}
domain (gcd) = (Z >< Z) \ { (0,0)}
GCD (m,n) = GCD (n,m)
GCD (m,0) = m
GCD (m,n) = GCD (m - n,n)
GCD (m,n) = GCD (m MOD n, n)
provided that the left hand side is defined. These properties give rise to the Euclidean algorithm for calculating GCD
GCD (m,0) = m
GCD (0,m) = m
GCD (m,n) = GCD (m MOD n, n) if n >= m > 0
GCD (m,n) = GCD (m, n MOD m) if m >= n > 0
Definition of "divides"
The symbol "|" means "divides" or "is a factor of". By definition, when a and b are integers a|b iff there exists an integer k such that ak=b. This definition leads to some unintuitive results, such as all integers (including zero) divide zero. By the way, "iff" means "if and only if".
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Number Theory Definitions and Principles
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Number Theory Definitions and Principles
Number Theory Definitions and Principles
Number Theory Definitions and Principles
Modulo Arithmetic
Division
Some properties of the Greatest Common Divisor (GCD)
Definition of "divides"
Definition of GCD, "Greatest Common Divisor"
Well-Ordering Principle
Division Algorithm Property
Linear Combination Property
Euclidean Algorithm Property, a.k.a. B�zout's Identity.
Euclid's principle:
Exercise: Proof
Other Properties of GCD
Internet references
|
d divides m and d divides n}
domain (gcd) = (Z >< Z) \ { (0,0)}
GCD (m,n) = GCD (n,m)
GCD (m,0) = m
GCD (m,n) = GCD (m - n,n)
GCD (m,n) = GCD (m MOD n, n)
provided that the left hand side is defined. These properties give rise to the Euclidean algorithm for calculating GCD
GCD (m,0) = m
GCD (0,m) = m
GCD (m,n) = GCD (m MOD n, n) if n >= m > 0
GCD (m,n) = GCD (m, n MOD m) if m >= n > 0
Definition of "divides"
The symbol "|" means "divides" or "is a factor of". By definition, when a and b are integers a|b iff there exists an integer k such that ak=b. This definition leads to some unintuitive results, such as all integers (including zero) divide zero. By the way, "iff" means "if and only if". Definition of GCD, "Greatest Common Divisor"
GCD (a,b)=c means c>=0, c|a and c|b. Furthermore, every integer d that divides a and b also divides c.
To put it another way, here are the properties of GCD, in a nutshell -- this form is useful for proofs, because it provides a checklist of things you must show in order to establish GCD (a,b) as well as a set of facts you know if you know GCD (a,b): GCD (a,b)>=0
GCD (a,b)|a
GCD (a,b)|b
if integer d exists such that d|a and d|b then d|GCD (a,b)
That last item in the checklist is equivalent to the following statement: for all integers d, it is true that d-|a or d-|b or d|GCD (a,b)
Cancellation Rule
m * p == m * q (mod n) and GCD (m,n) = 1 ==> p == q (mod n)
Thus m can safely be cancelled mod n so long as GCD (m,n) = 1. Prime Numbers
An integer p is a prime number if and only if
p > 1 and {x :
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Number Theory Definitions and Principles
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Number Theory Definitions and Principles
Number Theory Definitions and Principles
Number Theory Definitions and Principles
Modulo Arithmetic
Division
Some properties of the Greatest Common Divisor (GCD)
Definition of "divides"
Definition of GCD, "Greatest Common Divisor"
Well-Ordering Principle
Division Algorithm Property
Linear Combination Property
Euclidean Algorithm Property, a.k.a. B�zout's Identity.
Euclid's principle:
Exercise: Proof
Other Properties of GCD
Internet references
|
Definition of GCD, "Greatest Common Divisor"
GCD (a,b)=c means c>=0, c|a and c|b. Furthermore, every integer d that divides a and b also divides c.
To put it another way, here are the properties of GCD, in a nutshell -- this form is useful for proofs, because it provides a checklist of things you must show in order to establish GCD (a,b) as well as a set of facts you know if you know GCD (a,b): GCD (a,b)>=0
GCD (a,b)|a
GCD (a,b)|b
if integer d exists such that d|a and d|b then d|GCD (a,b)
That last item in the checklist is equivalent to the following statement: for all integers d, it is true that d-|a or d-|b or d|GCD (a,b)
Cancellation Rule
m * p == m * q (mod n) and GCD (m,n) = 1 ==> p == q (mod n)
Thus m can safely be cancelled mod n so long as GCD (m,n) = 1. Prime Numbers
An integer p is a prime number if and only if
p > 1 and {x : x divides p} = {1, p}
If p and q are both prime and p divides q, then p = q.
If p is prime and p does not divide n then GCD (p,n) = 1. If p is prime and p divides n*m then p must divide either n or m (or both). Any integer n can be factorized uniquely into the product of a list of primes p1 * p2 * ...
Well-Ordering Principle
...says that a set of positive integers has a smallest element. Let S be a nonempty set of positive integers. There exists some element a of S such that a<=b for all elements b of S.
Division Algorithm Property
If a and m are any integers with m not zero, then there are unique integers q and r such that a = qm+r with 0 < r < |m|.
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Number Theory Definitions and Principles
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Number Theory Definitions and Principles
Number Theory Definitions and Principles
Number Theory Definitions and Principles
Modulo Arithmetic
Division
Some properties of the Greatest Common Divisor (GCD)
Definition of "divides"
Definition of GCD, "Greatest Common Divisor"
Well-Ordering Principle
Division Algorithm Property
Linear Combination Property
Euclidean Algorithm Property, a.k.a. B�zout's Identity.
Euclid's principle:
Exercise: Proof
Other Properties of GCD
Internet references
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x divides p} = {1, p}
If p and q are both prime and p divides q, then p = q.
If p is prime and p does not divide n then GCD (p,n) = 1. If p is prime and p divides n*m then p must divide either n or m (or both). Any integer n can be factorized uniquely into the product of a list of primes p1 * p2 * ...
Well-Ordering Principle
...says that a set of positive integers has a smallest element. Let S be a nonempty set of positive integers. There exists some element a of S such that a<=b for all elements b of S.
Division Algorithm Property
If a and m are any integers with m not zero, then there are unique integers q and r such that a = qm+r with 0 < r < |m|. Linear Combination Property
If a|b and a|c then a| (bx + cy) for any integers x and y.
Proof: Suppose a|b and a|c. Then integers q and r exist such that b = aq and c = ar. So, for any integers x and y, bx + cy = a (qx + ry)
and qx + ry is an integer, and hence a| (bx + cy). An application of Linear Combination is this:
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Number Theory Definitions and Principles
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Number Theory Definitions and Principles
Number Theory Definitions and Principles
Number Theory Definitions and Principles
Modulo Arithmetic
Division
Some properties of the Greatest Common Divisor (GCD)
Definition of "divides"
Definition of GCD, "Greatest Common Divisor"
Well-Ordering Principle
Division Algorithm Property
Linear Combination Property
Euclidean Algorithm Property, a.k.a. B�zout's Identity.
Euclid's principle:
Exercise: Proof
Other Properties of GCD
Internet references
|
Linear Combination Property
If a|b and a|c then a| (bx + cy) for any integers x and y.
Proof: Suppose a|b and a|c. Then integers q and r exist such that b = aq and c = ar. So, for any integers x and y, bx + cy = a (qx + ry)
and qx + ry is an integer, and hence a| (bx + cy). An application of Linear Combination is this: GCD (m,n)=GCD (m+kn,n), where k is any integer. Proof: GCD (m+kn,n)|n and GCD (m+kn,n)|m+kn, by the definition of GCD. GCD (m+kn,n)| (m+kn)-kn, by the linear combination property
GCD (m+kn,n)|m
GCD (m+kn,n)|GCD (m,n)
similarly, GCD (m,n)|GCD (m+kn,n)
so GCD (m,n)=GCD (m+kn,n), where k is an integer. Euclidean Algorithm Property, a.k.a.
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Number Theory Definitions and Principles
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Number Theory Definitions and Principles
Number Theory Definitions and Principles
Number Theory Definitions and Principles
Modulo Arithmetic
Division
Some properties of the Greatest Common Divisor (GCD)
Definition of "divides"
Definition of GCD, "Greatest Common Divisor"
Well-Ordering Principle
Division Algorithm Property
Linear Combination Property
Euclidean Algorithm Property, a.k.a. B�zout's Identity.
Euclid's principle:
Exercise: Proof
Other Properties of GCD
Internet references
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GCD (m,n)=GCD (m+kn,n), where k is any integer. Proof: GCD (m+kn,n)|n and GCD (m+kn,n)|m+kn, by the definition of GCD. GCD (m+kn,n)| (m+kn)-kn, by the linear combination property
GCD (m+kn,n)|m
GCD (m+kn,n)|GCD (m,n)
similarly, GCD (m,n)|GCD (m+kn,n)
so GCD (m,n)=GCD (m+kn,n), where k is an integer. Euclidean Algorithm Property, a.k.a. B�zout's Identity. For any integers a and b, there exist integers x and y such that GCD (a,b)=ax+by. If a and b are zero, then x=y=0, and the property is obviously true. Now we can assume a or b is nonzero. Let S be the set of positive integers of the form sa + tb, where s and t are integers.
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Number Theory Definitions and Principles
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Number Theory Definitions and Principles
Number Theory Definitions and Principles
Number Theory Definitions and Principles
Modulo Arithmetic
Division
Some properties of the Greatest Common Divisor (GCD)
Definition of "divides"
Definition of GCD, "Greatest Common Divisor"
Well-Ordering Principle
Division Algorithm Property
Linear Combination Property
Euclidean Algorithm Property, a.k.a. B�zout's Identity.
Euclid's principle:
Exercise: Proof
Other Properties of GCD
Internet references
|
B�zout's Identity. For any integers a and b, there exist integers x and y such that GCD (a,b)=ax+by. If a and b are zero, then x=y=0, and the property is obviously true. Now we can assume a or b is nonzero. Let S be the set of positive integers of the form sa + tb, where s and t are integers. Set S isn't empty so it has a smallest element by the well-ordering principle. Let d=xa+yb be the smallest element of S.
Now I'll divide a by d to show that d|a. By the division algorithm, there are integers q and r such that a=dq+r, and 0<=r<d. r=a-dq
r=a- (xa+yb)q
r= (1-xq)a + (-yq)b,
which has the form of elements of S, so either r=0 or r is in S.
But r can't be in S, because r<d, the smallest element of S. So r=0. a=dq+r, so a=dq, so d|a.
| 5,228 | 5,968 |
msmarco_v2.1_doc_00_8093742#10_12501954
|
http://2000clicks.com/MathHelp/NumberTheoryDefinitions.aspx
|
Number Theory Definitions and Principles
|
Number Theory Definitions and Principles
Number Theory Definitions and Principles
Number Theory Definitions and Principles
Modulo Arithmetic
Division
Some properties of the Greatest Common Divisor (GCD)
Definition of "divides"
Definition of GCD, "Greatest Common Divisor"
Well-Ordering Principle
Division Algorithm Property
Linear Combination Property
Euclidean Algorithm Property, a.k.a. B�zout's Identity.
Euclid's principle:
Exercise: Proof
Other Properties of GCD
Internet references
|
Set S isn't empty so it has a smallest element by the well-ordering principle. Let d=xa+yb be the smallest element of S.
Now I'll divide a by d to show that d|a. By the division algorithm, there are integers q and r such that a=dq+r, and 0<=r<d. r=a-dq
r=a- (xa+yb)q
r= (1-xq)a + (-yq)b,
which has the form of elements of S, so either r=0 or r is in S.
But r can't be in S, because r<d, the smallest element of S. So r=0. a=dq+r, so a=dq, so d|a. Similarly (starting with the division algorithm) we can show d|b. Thus d is a divisor of a and b.
If c is an integer and c|a and c|b, then by the linear combination property c| (ax+by)
d=ax+by, so c|d. Any number that divides a and divides b also divides d, so d=GCD (a,b)
A constructive algorithm to find the integers x and y is the Extended Euclidean Algorithm. The converse: GCD (a,b)|ax+by is a consequence of the linear combination property.
| 5,521 | 6,415 |
msmarco_v2.1_doc_00_8093742#11_12503613
|
http://2000clicks.com/MathHelp/NumberTheoryDefinitions.aspx
|
Number Theory Definitions and Principles
|
Number Theory Definitions and Principles
Number Theory Definitions and Principles
Number Theory Definitions and Principles
Modulo Arithmetic
Division
Some properties of the Greatest Common Divisor (GCD)
Definition of "divides"
Definition of GCD, "Greatest Common Divisor"
Well-Ordering Principle
Division Algorithm Property
Linear Combination Property
Euclidean Algorithm Property, a.k.a. B�zout's Identity.
Euclid's principle:
Exercise: Proof
Other Properties of GCD
Internet references
|
Similarly (starting with the division algorithm) we can show d|b. Thus d is a divisor of a and b.
If c is an integer and c|a and c|b, then by the linear combination property c| (ax+by)
d=ax+by, so c|d. Any number that divides a and divides b also divides d, so d=GCD (a,b)
A constructive algorithm to find the integers x and y is the Extended Euclidean Algorithm. The converse: GCD (a,b)|ax+by is a consequence of the linear combination property. In particular, if x and y exist such that ax+by=1, then GCD (a,b)|1, so GCD (a,b)=1. Euclid's principle: If a prime p|ab then p|a or p|b, a and b integers. This is sometimes called Euclid's First Theorem. Proof using the Euclidean Algorithm Property:
| 5,968 | 6,666 |
msmarco_v2.1_doc_00_8093742#12_12505071
|
http://2000clicks.com/MathHelp/NumberTheoryDefinitions.aspx
|
Number Theory Definitions and Principles
|
Number Theory Definitions and Principles
Number Theory Definitions and Principles
Number Theory Definitions and Principles
Modulo Arithmetic
Division
Some properties of the Greatest Common Divisor (GCD)
Definition of "divides"
Definition of GCD, "Greatest Common Divisor"
Well-Ordering Principle
Division Algorithm Property
Linear Combination Property
Euclidean Algorithm Property, a.k.a. B�zout's Identity.
Euclid's principle:
Exercise: Proof
Other Properties of GCD
Internet references
|
In particular, if x and y exist such that ax+by=1, then GCD (a,b)|1, so GCD (a,b)=1. Euclid's principle: If a prime p|ab then p|a or p|b, a and b integers. This is sometimes called Euclid's First Theorem. Proof using the Euclidean Algorithm Property: Let us assume that p|ab. If p|a we're done. Now we can assume p-|a. In that case we have GCD (p,a)=1 since p is prime and hence there exist integers x and y such that 1=px+ay (Euclidean Algorithm Property). Multiply by b on both sides to get b=pxb+ayb.
| 6,415 | 6,919 |
msmarco_v2.1_doc_00_8093742#13_12506332
|
http://2000clicks.com/MathHelp/NumberTheoryDefinitions.aspx
|
Number Theory Definitions and Principles
|
Number Theory Definitions and Principles
Number Theory Definitions and Principles
Number Theory Definitions and Principles
Modulo Arithmetic
Division
Some properties of the Greatest Common Divisor (GCD)
Definition of "divides"
Definition of GCD, "Greatest Common Divisor"
Well-Ordering Principle
Division Algorithm Property
Linear Combination Property
Euclidean Algorithm Property, a.k.a. B�zout's Identity.
Euclid's principle:
Exercise: Proof
Other Properties of GCD
Internet references
|
Let us assume that p|ab. If p|a we're done. Now we can assume p-|a. In that case we have GCD (p,a)=1 since p is prime and hence there exist integers x and y such that 1=px+ay (Euclidean Algorithm Property). Multiply by b on both sides to get b=pxb+ayb. p|pxb and p|ayb (because p|ab),
so p| (pxb+ayb), by the Linear Combination Property. Thus p|b
Iff p and q are mutually prime, then p² and q² are mutually prime. If: Suppose p² and q² aren't mutually prime. GCD (p²,q²) is not 1, so there exists a prime r such that r|GCD (p²,q²)
r|p², so r|p by Euclid's principle
similarly, r|q², so r|q
r|GCD (p,q) because r|p and r|q
so GCD (p,q) isn't 1, a contradiction
Only if:
| 6,666 | 7,335 |
msmarco_v2.1_doc_00_8093742#14_12507815
|
http://2000clicks.com/MathHelp/NumberTheoryDefinitions.aspx
|
Number Theory Definitions and Principles
|
Number Theory Definitions and Principles
Number Theory Definitions and Principles
Number Theory Definitions and Principles
Modulo Arithmetic
Division
Some properties of the Greatest Common Divisor (GCD)
Definition of "divides"
Definition of GCD, "Greatest Common Divisor"
Well-Ordering Principle
Division Algorithm Property
Linear Combination Property
Euclidean Algorithm Property, a.k.a. B�zout's Identity.
Euclid's principle:
Exercise: Proof
Other Properties of GCD
Internet references
|
p|pxb and p|ayb (because p|ab),
so p| (pxb+ayb), by the Linear Combination Property. Thus p|b
Iff p and q are mutually prime, then p² and q² are mutually prime. If: Suppose p² and q² aren't mutually prime. GCD (p²,q²) is not 1, so there exists a prime r such that r|GCD (p²,q²)
r|p², so r|p by Euclid's principle
similarly, r|q², so r|q
r|GCD (p,q) because r|p and r|q
so GCD (p,q) isn't 1, a contradiction
Only if: Suppose p and q aren't mutually prime. Let r=GCD (p,q)
So r|p, which means r|p²
Also, r|q, which means r|q²
so r|GCD (p²,q²), a contradiction
Exercise: Proof
Suppose that a, b, c are integers such that GCD (a²,bc)=p where p is a prime. Prove that either a is coprime to b or a is coprime to c.
GCD (a²,bc)=p
p|a², (def. of GCD)
p|a, (Euclid's Principle)
An integer x exists such that px=a, (def.
| 6,919 | 7,731 |
msmarco_v2.1_doc_00_8093742#15_12509485
|
http://2000clicks.com/MathHelp/NumberTheoryDefinitions.aspx
|
Number Theory Definitions and Principles
|
Number Theory Definitions and Principles
Number Theory Definitions and Principles
Number Theory Definitions and Principles
Modulo Arithmetic
Division
Some properties of the Greatest Common Divisor (GCD)
Definition of "divides"
Definition of GCD, "Greatest Common Divisor"
Well-Ordering Principle
Division Algorithm Property
Linear Combination Property
Euclidean Algorithm Property, a.k.a. B�zout's Identity.
Euclid's principle:
Exercise: Proof
Other Properties of GCD
Internet references
|
Suppose p and q aren't mutually prime. Let r=GCD (p,q)
So r|p, which means r|p²
Also, r|q, which means r|q²
so r|GCD (p²,q²), a contradiction
Exercise: Proof
Suppose that a, b, c are integers such that GCD (a²,bc)=p where p is a prime. Prove that either a is coprime to b or a is coprime to c.
GCD (a²,bc)=p
p|a², (def. of GCD)
p|a, (Euclid's Principle)
An integer x exists such that px=a, (def. of "divides")
p²x²=a²
p²|a²
p|bc, (def. of GCD)
p|b or p|c (Euclid's Principle)
if p²|bc (and we know p²|a²) then p²|GCD (a²,bc), which is false because p=GCD (a²,bc) and p²-|p, so p²-|bc. if p|b and p|c then p²|bc, which is false, so either p-|b or p-|c. GCD (a,b)|a|a² and GCD (a,b)|b|bc, (def. of GCD)
so GCD (a,b)|GCD (a²,bc), (def.
| 7,335 | 8,068 |
msmarco_v2.1_doc_00_8093742#16_12511107
|
http://2000clicks.com/MathHelp/NumberTheoryDefinitions.aspx
|
Number Theory Definitions and Principles
|
Number Theory Definitions and Principles
Number Theory Definitions and Principles
Number Theory Definitions and Principles
Modulo Arithmetic
Division
Some properties of the Greatest Common Divisor (GCD)
Definition of "divides"
Definition of GCD, "Greatest Common Divisor"
Well-Ordering Principle
Division Algorithm Property
Linear Combination Property
Euclidean Algorithm Property, a.k.a. B�zout's Identity.
Euclid's principle:
Exercise: Proof
Other Properties of GCD
Internet references
|
of "divides")
p²x²=a²
p²|a²
p|bc, (def. of GCD)
p|b or p|c (Euclid's Principle)
if p²|bc (and we know p²|a²) then p²|GCD (a²,bc), which is false because p=GCD (a²,bc) and p²-|p, so p²-|bc. if p|b and p|c then p²|bc, which is false, so either p-|b or p-|c. GCD (a,b)|a|a² and GCD (a,b)|b|bc, (def. of GCD)
so GCD (a,b)|GCD (a²,bc), (def. of GCD (a²,bc))
so GCD (a,b)|p,
so either GCD (a,b)=1 or GCD (a,b)=p (def. of prime number)
Similarly, either GCD (a,c)=1 or GCD (a,c)=p. If both GCD (a,b)=p and GCD (a,c)=p then p|b and p|c, but we've already shown that p doesn't divide both b and c.
That means either GCD (a,b)=1 or GCD (a,c)=1. Other Properties of GCD
Equality Property of Division: If a and b are nonnegative integers, then a|b and b|a ==> a=b.
| 7,732 | 8,484 |
msmarco_v2.1_doc_00_8093742#17_12512715
|
http://2000clicks.com/MathHelp/NumberTheoryDefinitions.aspx
|
Number Theory Definitions and Principles
|
Number Theory Definitions and Principles
Number Theory Definitions and Principles
Number Theory Definitions and Principles
Modulo Arithmetic
Division
Some properties of the Greatest Common Divisor (GCD)
Definition of "divides"
Definition of GCD, "Greatest Common Divisor"
Well-Ordering Principle
Division Algorithm Property
Linear Combination Property
Euclidean Algorithm Property, a.k.a. B�zout's Identity.
Euclid's principle:
Exercise: Proof
Other Properties of GCD
Internet references
|
of GCD (a²,bc))
so GCD (a,b)|p,
so either GCD (a,b)=1 or GCD (a,b)=p (def. of prime number)
Similarly, either GCD (a,c)=1 or GCD (a,c)=p. If both GCD (a,b)=p and GCD (a,c)=p then p|b and p|c, but we've already shown that p doesn't divide both b and c.
That means either GCD (a,b)=1 or GCD (a,c)=1. Other Properties of GCD
Equality Property of Division: If a and b are nonnegative integers, then a|b and b|a ==> a=b. Proof: Integers n and m exist such that an=b, and bm=a. If a=0 then an=0=b. Likewise, if b=0 then bm=0=a,
so it's true when a or b is zero. From here on, we can assume a and b are both positive...
Substituting an in place of b, anm=a, and since a is not zero, nm=1.
| 8,069 | 8,750 |
msmarco_v2.1_doc_00_8093742#18_12514166
|
http://2000clicks.com/MathHelp/NumberTheoryDefinitions.aspx
|
Number Theory Definitions and Principles
|
Number Theory Definitions and Principles
Number Theory Definitions and Principles
Number Theory Definitions and Principles
Modulo Arithmetic
Division
Some properties of the Greatest Common Divisor (GCD)
Definition of "divides"
Definition of GCD, "Greatest Common Divisor"
Well-Ordering Principle
Division Algorithm Property
Linear Combination Property
Euclidean Algorithm Property, a.k.a. B�zout's Identity.
Euclid's principle:
Exercise: Proof
Other Properties of GCD
Internet references
|
Proof: Integers n and m exist such that an=b, and bm=a. If a=0 then an=0=b. Likewise, if b=0 then bm=0=a,
so it's true when a or b is zero. From here on, we can assume a and b are both positive...
Substituting an in place of b, anm=a, and since a is not zero, nm=1. Both n and m are positive, because n=b/a>0 and m=a/b>0. Solving n>=1, m>=1, and nm=1, we see that n=1 and m=1. Since an=b, and n=1, we have proved that a=b. Property 2. If a = bt + r, for integers t and r, then GCD (a,b) = GCD (b,r).
| 8,484 | 8,984 |
msmarco_v2.1_doc_00_8093742#19_12515425
|
http://2000clicks.com/MathHelp/NumberTheoryDefinitions.aspx
|
Number Theory Definitions and Principles
|
Number Theory Definitions and Principles
Number Theory Definitions and Principles
Number Theory Definitions and Principles
Modulo Arithmetic
Division
Some properties of the Greatest Common Divisor (GCD)
Definition of "divides"
Definition of GCD, "Greatest Common Divisor"
Well-Ordering Principle
Division Algorithm Property
Linear Combination Property
Euclidean Algorithm Property, a.k.a. B�zout's Identity.
Euclid's principle:
Exercise: Proof
Other Properties of GCD
Internet references
|
Both n and m are positive, because n=b/a>0 and m=a/b>0. Solving n>=1, m>=1, and nm=1, we see that n=1 and m=1. Since an=b, and n=1, we have proved that a=b. Property 2. If a = bt + r, for integers t and r, then GCD (a,b) = GCD (b,r). Proof: Every common divisor of a and b also divides r, because r=a-bt is a linear combination of a and b.
So GCD (a,b)|r, and since GCD (a,b)|b, it is true that GCD (a,b) is a common divisor of b and r.
By the same logic, every common divisor of b and r also divides a, because a=bt+r, a linear combination of b and r.
GCD (b,r)|a, and since GCD (b,r)|b, it is true that GCD (b,r) is a common divisor of a and b.
The product of any two coprimes of n is another coprime of n
That is, if GCD (a,n)=1 and GCD (b,n)=1 then GCD (ab,n)=1. Here's the proof . This fact, and the fact that follows, are used to show that the equivalence classes (mod n) of the coprimes of n form a cyclic group under multiplication. If a, b, c are coprimes of n, and b is not equal to c, then ab is not equal to ac
That is, the following five statements can't all be true:
| 8,750 | 9,831 |
msmarco_v2.1_doc_00_8093742#20_12517269
|
http://2000clicks.com/MathHelp/NumberTheoryDefinitions.aspx
|
Number Theory Definitions and Principles
|
Number Theory Definitions and Principles
Number Theory Definitions and Principles
Number Theory Definitions and Principles
Modulo Arithmetic
Division
Some properties of the Greatest Common Divisor (GCD)
Definition of "divides"
Definition of GCD, "Greatest Common Divisor"
Well-Ordering Principle
Division Algorithm Property
Linear Combination Property
Euclidean Algorithm Property, a.k.a. B�zout's Identity.
Euclid's principle:
Exercise: Proof
Other Properties of GCD
Internet references
|
Proof: Every common divisor of a and b also divides r, because r=a-bt is a linear combination of a and b.
So GCD (a,b)|r, and since GCD (a,b)|b, it is true that GCD (a,b) is a common divisor of b and r.
By the same logic, every common divisor of b and r also divides a, because a=bt+r, a linear combination of b and r.
GCD (b,r)|a, and since GCD (b,r)|b, it is true that GCD (b,r) is a common divisor of a and b.
The product of any two coprimes of n is another coprime of n
That is, if GCD (a,n)=1 and GCD (b,n)=1 then GCD (ab,n)=1. Here's the proof . This fact, and the fact that follows, are used to show that the equivalence classes (mod n) of the coprimes of n form a cyclic group under multiplication. If a, b, c are coprimes of n, and b is not equal to c, then ab is not equal to ac
That is, the following five statements can't all be true: 1. GCD (a,n)=1,
2. GCD (b,n)=1,
3. GCD (c,n)=1,
4. a≠b
5.
| 8,984 | 9,889 |
msmarco_v2.1_doc_00_8093742#21_12518946
|
http://2000clicks.com/MathHelp/NumberTheoryDefinitions.aspx
|
Number Theory Definitions and Principles
|
Number Theory Definitions and Principles
Number Theory Definitions and Principles
Number Theory Definitions and Principles
Modulo Arithmetic
Division
Some properties of the Greatest Common Divisor (GCD)
Definition of "divides"
Definition of GCD, "Greatest Common Divisor"
Well-Ordering Principle
Division Algorithm Property
Linear Combination Property
Euclidean Algorithm Property, a.k.a. B�zout's Identity.
Euclid's principle:
Exercise: Proof
Other Properties of GCD
Internet references
|
1. GCD (a,n)=1,
2. GCD (b,n)=1,
3. GCD (c,n)=1,
4. a≠b
5. ab=ac
Here's the proof. Internet references
B�zout's Identity, from Mathworld, which says: For any integers a
| 9,831 | 9,999 |
msmarco_v2.1_doc_00_8105801#0_12519887
|
http://2000clicks.com/graeme/LangNumbersMillionsAndBillionsAndZillions.htm
|
Millions and Billions and Zillions -- how high can you count?
|
Internet References
|
Millions and Billions and Zillions -- how high can you count? Millions and Billions and Zillions -- how high can you count? That depends on whether you know the words for very big numbers, and also two important factors: how fast can you count, and how much time to you have to spend on this crusade. Do you know the words for very big numbers? The words you use for big numbers depend on whether you live in the United States or in the United Kingdom (although someone emailed me to say the Brits have thrown in the towel, and adopted the Yankee method. If that's true, then read: old UK). Here is a table that might help you: Power of 10
Number
American Name
British Name
(not used much any more)
3
1, 000
Thousand
Thousand
6
1, 000, 000
Million
Million
9
1, 000, 000, 000
Billion
Thousand Million, or "Milliard"
12
1, 000, 000, 000, 000
Trillion
Billion
15
1, 000, 000, 000, 000, 000
Quadrillion
Thousand Billion
18
1, 000, 000, 000, 000, 000, 000
Quintillion
Trillion
21
1, 000, 000, 000, 000, 000, 000, 000
Sextillion
Thousand Trillion
24
1, 000, 000, 000, 000, 000, 000, 000, 000
Septillion
Quadrillion
27
1, 000, 000, 000, 000, 000, 000, 000, 000, 000
Octillion
Thousand Quadrillion
30
1, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000
Nonillion
Sexillion
The British Names comprise a better system of naming large numbers, because it doesn't use up the Latin names (tri-, quad-, etc.)
| 0 | 1,396 |
msmarco_v2.1_doc_00_8105801#1_12521627
|
http://2000clicks.com/graeme/LangNumbersMillionsAndBillionsAndZillions.htm
|
Millions and Billions and Zillions -- how high can you count?
|
Internet References
|
The words you use for big numbers depend on whether you live in the United States or in the United Kingdom (although someone emailed me to say the Brits have thrown in the towel, and adopted the Yankee method. If that's true, then read: old UK). Here is a table that might help you: Power of 10
Number
American Name
British Name
(not used much any more)
3
1, 000
Thousand
Thousand
6
1, 000, 000
Million
Million
9
1, 000, 000, 000
Billion
Thousand Million, or "Milliard"
12
1, 000, 000, 000, 000
Trillion
Billion
15
1, 000, 000, 000, 000, 000
Quadrillion
Thousand Billion
18
1, 000, 000, 000, 000, 000, 000
Quintillion
Trillion
21
1, 000, 000, 000, 000, 000, 000, 000
Sextillion
Thousand Trillion
24
1, 000, 000, 000, 000, 000, 000, 000, 000
Septillion
Quadrillion
27
1, 000, 000, 000, 000, 000, 000, 000, 000, 000
Octillion
Thousand Quadrillion
30
1, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000
Nonillion
Sexillion
The British Names comprise a better system of naming large numbers, because it doesn't use up the Latin names (tri-, quad-, etc.) so fast. On the other hand, the American system is better for medium sized numbers like 1,335,168,736,797,130,752. In America, this number is
one quintillion,
three hundred thirty-five quadrillion,
one hundred sixty-eight trillion,
seven hundred thirty-six billion,
seven hundred ninety-seven million,
one hundred thirty thousand,
seven hundred fifty-two. Using the old UK system, 1,335,168,736,797,130,752 is
one trillion,
three hundred thirty-five thousand billion,
one hundred sixty-eight billion,
seven hundred thirty-six thousand million,
seven hundred ninety-seven million,
one hundred thirty thousand,
seven hundred fifty-two. For medium-sized numbers like this that have many significant digits, the old British system is downright confusing because thousands and millions can modify trillions resulting in gazillions!
| 346 | 2,223 |
msmarco_v2.1_doc_00_8105801#2_12523865
|
http://2000clicks.com/graeme/LangNumbersMillionsAndBillionsAndZillions.htm
|
Millions and Billions and Zillions -- how high can you count?
|
Internet References
|
so fast. On the other hand, the American system is better for medium sized numbers like 1,335,168,736,797,130,752. In America, this number is
one quintillion,
three hundred thirty-five quadrillion,
one hundred sixty-eight trillion,
seven hundred thirty-six billion,
seven hundred ninety-seven million,
one hundred thirty thousand,
seven hundred fifty-two. Using the old UK system, 1,335,168,736,797,130,752 is
one trillion,
three hundred thirty-five thousand billion,
one hundred sixty-eight billion,
seven hundred thirty-six thousand million,
seven hundred ninety-seven million,
one hundred thirty thousand,
seven hundred fifty-two. For medium-sized numbers like this that have many significant digits, the old British system is downright confusing because thousands and millions can modify trillions resulting in gazillions! That's enough. Click the Back button on your browser now. Internet References
" Large Numbers ", from Mathworld
| 1,397 | 2,336 |
msmarco_v2.1_doc_00_8108459#0_12525122
|
http://2001-2009.state.gov/e/eeb/ifd/2008/101774.htm
|
West Bank/Gaza
|
West Bank/Gaza
West Bank/Gaza
|
West Bank/Gaza
West Bank/Gaza
2008 Investment Climate Statement – West Bank/Gaza
Introduction
Internal Palestinian political issues and the Israeli-Palestinian conflict continue to impact negatively the development of the Palestinian economy in the West Bank and Gaza Strip (WB/G). Following the establishment of a Palestinian Authority (PA) Government under the leadership of Prime Minister Salam Fayyad in June 2007, the PA has demonstrated a renewed determination to improve the investment climate in the WB/G and to attract foreign investment. The PA has undertaken a number of significant reforms and prepared a three year reform and development plan that was endorsed by the international community in December 2007. The PA's development plan emphasizes the importance of private sector investment and growth as a vital source of new jobs and a sustainable economy. At the time this report was drafted, Hamas, a designated Foreign Terrorist Organization (FTO), remained in control of the Gaza Strip, having violently seized power in June 2007. Where applicable, this report addresses issues related to investment in the Gaza Strip, although there are currently no opportunities for meaningful private investment in Gaza due to Hamas' control. This report focuses on investment issues related to areas under the administrative jurisdiction of the Palestinian Authority, except where explicitly stated. Given the changing circumstances on the ground, potential investors are encouraged to contact the U.S. Consulate General in Jerusalem and the Foreign Commercial Service for the latest information. Openness To Foreign Investment
PA laws have established a legal structure that aims to promote foreign investment. The 1998 Investment Law guarantees the repatriation of foreign capital and prohibits expropriation and nationalization of approved foreign investments.
| 0 | 1,899 |
msmarco_v2.1_doc_00_8108459#1_12527298
|
http://2001-2009.state.gov/e/eeb/ifd/2008/101774.htm
|
West Bank/Gaza
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West Bank/Gaza
West Bank/Gaza
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Where applicable, this report addresses issues related to investment in the Gaza Strip, although there are currently no opportunities for meaningful private investment in Gaza due to Hamas' control. This report focuses on investment issues related to areas under the administrative jurisdiction of the Palestinian Authority, except where explicitly stated. Given the changing circumstances on the ground, potential investors are encouraged to contact the U.S. Consulate General in Jerusalem and the Foreign Commercial Service for the latest information. Openness To Foreign Investment
PA laws have established a legal structure that aims to promote foreign investment. The 1998 Investment Law guarantees the repatriation of foreign capital and prohibits expropriation and nationalization of approved foreign investments. PA law states that no restrictions govern foreign currency accounts. Foreign Taxes: All new foreign investment in WB/G must be registered with the PA and approved by the relevant ministry/ministries. The income tax law is intended to incorporate both West Bank and Gaza. Ministry of Finance officials stated to USG officials in January 2008 that the PA's corporate tax rate is 15 percent, while personal income tax is specified according to the following:
| 1,079 | 2,355 |
msmarco_v2.1_doc_00_8108459#2_12528816
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http://2001-2009.state.gov/e/eeb/ifd/2008/101774.htm
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West Bank/Gaza
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West Bank/Gaza
West Bank/Gaza
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PA law states that no restrictions govern foreign currency accounts. Foreign Taxes: All new foreign investment in WB/G must be registered with the PA and approved by the relevant ministry/ministries. The income tax law is intended to incorporate both West Bank and Gaza. Ministry of Finance officials stated to USG officials in January 2008 that the PA's corporate tax rate is 15 percent, while personal income tax is specified according to the following: 5 percent for income between NIS 1 - 10,000; 10 percent for income between NIS 10,001 - 25,000; and
15 percent for all incomes above NIS 25,001. A 20 percent tax is withheld at source from dividends distributed in WB/G to shareholders of a foreign company. There are no taxes due on dividends distributed to shareholders of Palestinian companies regardless of where they live or their nationality, and regardless of whether they are an individual or a company.
| 1,900 | 2,816 |
msmarco_v2.1_doc_00_8108459#3_12529974
|
http://2001-2009.state.gov/e/eeb/ifd/2008/101774.htm
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West Bank/Gaza
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West Bank/Gaza
West Bank/Gaza
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5 percent for income between NIS 1 - 10,000; 10 percent for income between NIS 10,001 - 25,000; and
15 percent for all incomes above NIS 25,001. A 20 percent tax is withheld at source from dividends distributed in WB/G to shareholders of a foreign company. There are no taxes due on dividends distributed to shareholders of Palestinian companies regardless of where they live or their nationality, and regardless of whether they are an individual or a company. An automatic deduction at the source of 25 percent is withheld from companies, unless they obtain a "Deduction at the Source Certificate," which grants a reduced rate that ranges between zero and five percent. Applications for these certificates are available from district tax offices. Exemptions: The 1998 Investment Law provides a number of incentives, including exemption from taxes, for PA-approved domestic and foreign investment. To benefit from these incentives, investors must apply to the Palestinian Investment Promotion Agency (PIPA), a department of the PA Ministry of National Economy, and present it with a completed investment application and feasibility study.
| 2,355 | 3,494 |
msmarco_v2.1_doc_00_8108459#4_12531356
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http://2001-2009.state.gov/e/eeb/ifd/2008/101774.htm
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West Bank/Gaza
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West Bank/Gaza
West Bank/Gaza
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An automatic deduction at the source of 25 percent is withheld from companies, unless they obtain a "Deduction at the Source Certificate," which grants a reduced rate that ranges between zero and five percent. Applications for these certificates are available from district tax offices. Exemptions: The 1998 Investment Law provides a number of incentives, including exemption from taxes, for PA-approved domestic and foreign investment. To benefit from these incentives, investors must apply to the Palestinian Investment Promotion Agency (PIPA), a department of the PA Ministry of National Economy, and present it with a completed investment application and feasibility study. PIPA is composed of both public and private sector members. Pre-approval: Certain investment categories require PA Council of Ministers' pre-approval. These include investments involving (1) weapons and ammunition, (2) aviation products and airport construction, (3) electrical power generation/distribution, (4) reprocessing of petroleum and its derivatives, (5) waste and solid waste reprocessing, (6) wired and wireless telecommunication, and (7) radio and television. Conversion And Transfer Policies
The 1998 Investment Law guarantees investors the repatriation of all financial resources, including capital, profits, dividends, wages, and salaries.
| 2,817 | 4,149 |
msmarco_v2.1_doc_00_8108459#5_12532932
|
http://2001-2009.state.gov/e/eeb/ifd/2008/101774.htm
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West Bank/Gaza
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West Bank/Gaza
West Bank/Gaza
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PIPA is composed of both public and private sector members. Pre-approval: Certain investment categories require PA Council of Ministers' pre-approval. These include investments involving (1) weapons and ammunition, (2) aviation products and airport construction, (3) electrical power generation/distribution, (4) reprocessing of petroleum and its derivatives, (5) waste and solid waste reprocessing, (6) wired and wireless telecommunication, and (7) radio and television. Conversion And Transfer Policies
The 1998 Investment Law guarantees investors the repatriation of all financial resources, including capital, profits, dividends, wages, and salaries. There are no other PA restrictions governing foreign currency accounts and currency transfer policies. Expropriation And Compensation
The 1998 Investment Law prohibits expropriation and nationalization of approved foreign investments, except for the pursuit of the public good, which shall be in return for fair compensation based on market prices and for losses suffered because of such expropriation. PA sources and independent lawyers say that any Palestinian citizen can file a petition or a lawsuit against the PA. There are on-going court cases involving illegal confiscation of property by PA senior government officials; however, there has been no ruling on most of these cases.
| 3,495 | 4,836 |
msmarco_v2.1_doc_00_8108459#6_12534516
|
http://2001-2009.state.gov/e/eeb/ifd/2008/101774.htm
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West Bank/Gaza
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West Bank/Gaza
West Bank/Gaza
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There are no other PA restrictions governing foreign currency accounts and currency transfer policies. Expropriation And Compensation
The 1998 Investment Law prohibits expropriation and nationalization of approved foreign investments, except for the pursuit of the public good, which shall be in return for fair compensation based on market prices and for losses suffered because of such expropriation. PA sources and independent lawyers say that any Palestinian citizen can file a petition or a lawsuit against the PA. There are on-going court cases involving illegal confiscation of property by PA senior government officials; however, there has been no ruling on most of these cases. A general lack of confidence in the judicial system has prompted citizens to look for alternative means of arbitration to resolve such disputes. Dispute Settlement
The 1998 Investment Law provides for dispute resolution between the investor and official agencies by binding independent arbitration or in Palestinian courts. It has been reported that some contracts contain clauses referring dispute resolutions to the London Court of Arbitration. Performance Requirement And Incentives
Certain incentives apply to Palestinian Investment Promotion Agency-approved investments: Investments whose value is between USD 100,000 and USD 1 million will be exempt from income tax for five years and be subject to income tax on their net profit at 10 percent for an additional eight years;
| 4,150 | 5,617 |
msmarco_v2.1_doc_00_8108459#7_12536227
|
http://2001-2009.state.gov/e/eeb/ifd/2008/101774.htm
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West Bank/Gaza
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West Bank/Gaza
West Bank/Gaza
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A general lack of confidence in the judicial system has prompted citizens to look for alternative means of arbitration to resolve such disputes. Dispute Settlement
The 1998 Investment Law provides for dispute resolution between the investor and official agencies by binding independent arbitration or in Palestinian courts. It has been reported that some contracts contain clauses referring dispute resolutions to the London Court of Arbitration. Performance Requirement And Incentives
Certain incentives apply to Palestinian Investment Promotion Agency-approved investments: Investments whose value is between USD 100,000 and USD 1 million will be exempt from income tax for five years and be subject to income tax on their net profit at 10 percent for an additional eight years; Investments whose value is between USD 1 million and USD 5 million will be exempt from income tax for five years and be subject to income tax on their net profit at 10 percent for an additional 12 years; Investments whose value is USD 5 million and above will be exempt from income tax for five years and be subject to income tax on their net profit at 10 percent for an additional 16 years; Special projects recommended by PIPA and approved by the Council of Ministers will be exempted from income tax for five years and be subject to income tax on their net profit at 10 percent for an additional 20 years; and
Investments in information technology (IT) training may be capitalized and depreciated for tax purposes. The United States continues to support private sector development in the WB/G. In 2007, the Overseas Private Investment Corporation (OPIC), working with Palestinian and U.S. partners, helped establish a program that will generate at least USD 228 million in lending to Palestinian small and medium enterprises over the next 10 years.
| 4,837 | 6,669 |
msmarco_v2.1_doc_00_8108459#8_12538303
|
http://2001-2009.state.gov/e/eeb/ifd/2008/101774.htm
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West Bank/Gaza
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West Bank/Gaza
West Bank/Gaza
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Investments whose value is between USD 1 million and USD 5 million will be exempt from income tax for five years and be subject to income tax on their net profit at 10 percent for an additional 12 years; Investments whose value is USD 5 million and above will be exempt from income tax for five years and be subject to income tax on their net profit at 10 percent for an additional 16 years; Special projects recommended by PIPA and approved by the Council of Ministers will be exempted from income tax for five years and be subject to income tax on their net profit at 10 percent for an additional 20 years; and
Investments in information technology (IT) training may be capitalized and depreciated for tax purposes. The United States continues to support private sector development in the WB/G. In 2007, the Overseas Private Investment Corporation (OPIC), working with Palestinian and U.S. partners, helped establish a program that will generate at least USD 228 million in lending to Palestinian small and medium enterprises over the next 10 years. OPIC is also seeking to establish a new mortgage finance company which will offer long-term mortgage loans to potential home-buyers. By dramatically expanding access to long-term credit, the mortgage facility will support several new affordable housing development projects, thus stimulating the construction sector. OPIC is also investigating the possibility of providing political risk/trade disruption insurance to businesses operating in the West Bank. In addition, in December 2007, the Secretary of State launched the U.S.-Palestinian Public-Private Partnership to support the development of economic and educational opportunities for Palestinian youth and to foster business opportunities in the West Bank. Right To Private Ownership And Establishment
The right to private ownership in Gaza is guaranteed by British Mandate law, as amended by regulations issued by the PA.
| 5,617 | 7,549 |
msmarco_v2.1_doc_00_8108459#9_12540477
|
http://2001-2009.state.gov/e/eeb/ifd/2008/101774.htm
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West Bank/Gaza
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West Bank/Gaza
West Bank/Gaza
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OPIC is also seeking to establish a new mortgage finance company which will offer long-term mortgage loans to potential home-buyers. By dramatically expanding access to long-term credit, the mortgage facility will support several new affordable housing development projects, thus stimulating the construction sector. OPIC is also investigating the possibility of providing political risk/trade disruption insurance to businesses operating in the West Bank. In addition, in December 2007, the Secretary of State launched the U.S.-Palestinian Public-Private Partnership to support the development of economic and educational opportunities for Palestinian youth and to foster business opportunities in the West Bank. Right To Private Ownership And Establishment
The right to private ownership in Gaza is guaranteed by British Mandate law, as amended by regulations issued by the PA. Jordanian law in the West Bank, as amended by PA regulations, similarly guarantees the right to private ownership. Foreigners must obtain permission from the PA before purchasing property in areas under PA civil authority and from the appropriate Israeli authorities before purchasing property in West Bank areas under Israeli control. PIPA outlines the following concerning foreign ownership of property: The Acquisition Law in the West Bank, which regulates foreign acquisition and the rental or lease of immovable properties, classifies foreigners into three categories: Foreigners who formerly possessed Palestinian or Jordanian passports shall have the right to own certain properties sufficient to erect buildings and/or for their agricultural projects.
| 6,670 | 8,309 |
msmarco_v2.1_doc_00_8108459#10_12542358
|
http://2001-2009.state.gov/e/eeb/ifd/2008/101774.htm
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West Bank/Gaza
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West Bank/Gaza
West Bank/Gaza
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Jordanian law in the West Bank, as amended by PA regulations, similarly guarantees the right to private ownership. Foreigners must obtain permission from the PA before purchasing property in areas under PA civil authority and from the appropriate Israeli authorities before purchasing property in West Bank areas under Israeli control. PIPA outlines the following concerning foreign ownership of property: The Acquisition Law in the West Bank, which regulates foreign acquisition and the rental or lease of immovable properties, classifies foreigners into three categories: Foreigners who formerly possessed Palestinian or Jordanian passports shall have the right to own certain properties sufficient to erect buildings and/or for their agricultural projects. Foreigners who hold other Arab nationality passports have the right to own certain property that suffices for their living and business needs only. Other foreigners must receive permission from the PA Cabinet to own buildings or purchase land. It is critical that potential purchasers of land or buildings perform a title search to be assured that no outstanding violations or unpaid penalties exist on the property. Under current law, violations and penalties are transferred to the new owner. Accurate title search can only be obtained from the PA Land Authority (al-Taboh).
| 7,550 | 8,886 |
msmarco_v2.1_doc_00_8108459#11_12543936
|
http://2001-2009.state.gov/e/eeb/ifd/2008/101774.htm
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West Bank/Gaza
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West Bank/Gaza
West Bank/Gaza
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Foreigners who hold other Arab nationality passports have the right to own certain property that suffices for their living and business needs only. Other foreigners must receive permission from the PA Cabinet to own buildings or purchase land. It is critical that potential purchasers of land or buildings perform a title search to be assured that no outstanding violations or unpaid penalties exist on the property. Under current law, violations and penalties are transferred to the new owner. Accurate title search can only be obtained from the PA Land Authority (al-Taboh). Land registration is done through the Land Registries in Hebron, Ramallah, Qalquilya, Tulkarem, Nablus, Bethlehem, Jericho, Jenin, and Gaza City. In order to purchase land in WB/G, an application that includes supporting documents, such as deeds to the property and powers of attorney, should be submitted to the land registry office having jurisdiction over the land. Protection Of Property Rights
The West Bank and Gaza do not have a modern intellectual property rights (IPR) regime in place. The PA was indirectly committed to the GATT-TRIPS agreement when it signed the Interim Agreement on WB/Gaza according to Annex III (Protocol Concerning Civil Affairs), Appendix 1, Article 23. All IPR legislation pertaining to WB/G originates from British Mandate Law regardless of the change in control over the years.
| 8,309 | 9,700 |
msmarco_v2.1_doc_00_8108459#12_12545569
|
http://2001-2009.state.gov/e/eeb/ifd/2008/101774.htm
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West Bank/Gaza
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West Bank/Gaza
West Bank/Gaza
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Land registration is done through the Land Registries in Hebron, Ramallah, Qalquilya, Tulkarem, Nablus, Bethlehem, Jericho, Jenin, and Gaza City. In order to purchase land in WB/G, an application that includes supporting documents, such as deeds to the property and powers of attorney, should be submitted to the land registry office having jurisdiction over the land. Protection Of Property Rights
The West Bank and Gaza do not have a modern intellectual property rights (IPR) regime in place. The PA was indirectly committed to the GATT-TRIPS agreement when it signed the Interim Agreement on WB/Gaza according to Annex III (Protocol Concerning Civil Affairs), Appendix 1, Article 23. All IPR legislation pertaining to WB/G originates from British Mandate Law regardless of the change in control over the years. Pre-1967 era Jordanian laws concerning trademarks, patents, and designs are applicable in the West Bank. In Gaza, the Palestinian Trademark and Patent Laws of 1938, adopted during the British Mandate, are applicable. Registration under the two laws is very similar, and, despite different authorizin
| 8,887 | 10,000 |
msmarco_v2.1_doc_00_8128842#0_12546926
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http://2001-2009.state.gov/g/oes/continentalshelf/
|
Defining the Limits of the U.S. Continental Shelf
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Defining the Limits of the U.S. Continental Shelf
Defining the Limits of the U.S. Continental Shelf
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Defining the Limits of the U.S. Continental Shelf
Defining the Limits of the U.S. Continental Shelf
Since 2001, the United States has been engaged in gathering and analyzing data to determine the outer limits of its extended continental shelf (ECS). Under the Convention on the Law of the Sea, every coastal State has a continental shelf out to 200 nautical miles from its coastal baselines (or out to a maritime boundary with another coastal State), and beyond that distance if certain criteria are met. Article 76 of the Convention sets forth the criteria upon which a coastal State may determine a continental shelf that extends beyond 200 nautical miles. The ECS is that portion of the continental shelf that lies beyond this 200 nautical mile limit. Beginning in 2007 the effort to delimit the U.S. ECS became the Extended Continental Shelf Project, directed by an interagency task force. Defining the U.S. Extended Continental Shelf
The process to determine the outer limits of a State’s ECS requires the collection and analysis of data that describe the depth, shape, and geophysical characteristics of the seabed and sub-sea floor. Particularly important is bathymetric and sediment thickness data. The U.S. Extended Continental Shelf Task Force, an interagency body headed by the U.S. Department of State, coordinates the work to define the limits of the U.S. continental shelf. Participants in this Task Force include: State Department, NOAA (National Oceanic and Atmospheric Administration), the U.S. Geological Survey, the Executive Office of the President, the Joint Chiefs of Staff, the U.S. Navy, the U.S. Coast Guard, the Department of Energy, the National Science Foundation, the Environmental Protection Agency, the Minerals Management Service, and the Arctic Research Commission.
| 0 | 1,839 |
msmarco_v2.1_doc_00_8128842#1_12549155
|
http://2001-2009.state.gov/g/oes/continentalshelf/
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Defining the Limits of the U.S. Continental Shelf
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Defining the Limits of the U.S. Continental Shelf
Defining the Limits of the U.S. Continental Shelf
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Defining the U.S. Extended Continental Shelf
The process to determine the outer limits of a State’s ECS requires the collection and analysis of data that describe the depth, shape, and geophysical characteristics of the seabed and sub-sea floor. Particularly important is bathymetric and sediment thickness data. The U.S. Extended Continental Shelf Task Force, an interagency body headed by the U.S. Department of State, coordinates the work to define the limits of the U.S. continental shelf. Participants in this Task Force include: State Department, NOAA (National Oceanic and Atmospheric Administration), the U.S. Geological Survey, the Executive Office of the President, the Joint Chiefs of Staff, the U.S. Navy, the U.S. Coast Guard, the Department of Energy, the National Science Foundation, the Environmental Protection Agency, the Minerals Management Service, and the Arctic Research Commission. Why define the U.S. extended continental shelf? A coastal State can exercise certain sovereign rights over its continental shelf, including: exploration, exploitation, conservation, and management of non-living resources of the seabed and subsoil of the continental shelf, such as ferromanganese crusts, ferromanganese nodules, gas hydrate deposits, and petroleum; and exploration, exploitation, conservation, and management of living, "sedentary" resources, such as clams, crabs, scallops, sponges, and mollusks. While a continental shelf is coincident with the exclusive economic zone (EEZ) out to 200 nautical miles, the ECS is not an extension of the EEZ.
| 934 | 2,499 |
msmarco_v2.1_doc_00_8128842#2_12551070
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http://2001-2009.state.gov/g/oes/continentalshelf/
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Defining the Limits of the U.S. Continental Shelf
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Defining the Limits of the U.S. Continental Shelf
Defining the Limits of the U.S. Continental Shelf
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Why define the U.S. extended continental shelf? A coastal State can exercise certain sovereign rights over its continental shelf, including: exploration, exploitation, conservation, and management of non-living resources of the seabed and subsoil of the continental shelf, such as ferromanganese crusts, ferromanganese nodules, gas hydrate deposits, and petroleum; and exploration, exploitation, conservation, and management of living, "sedentary" resources, such as clams, crabs, scallops, sponges, and mollusks. While a continental shelf is coincident with the exclusive economic zone (EEZ) out to 200 nautical miles, the ECS is not an extension of the EEZ. Sovereign rights that apply to the EEZ, especially rights to the resources of the water column (e.g., pelagic fisheries) do not apply to the ECS. Establishing ECS limits will define the U.S. continental shelf in concrete geographical terms. Moreover, the United States has an inherent national interest in knowing, and declaring to others with specificity and certainty, the extent of sovereign rights with regard to the U.S. continental shelf beyond 200 nautical miles. Such certainty and international recognition is important to establishing the stability necessary for development and conservation of these potentially resource-rich areas. The collection and analysis of the data necessary to support the establishment of the U.S. ECS will, in itself, serve a range of other environmental, geologic, engineering, and resource management needs.
| 1,839 | 3,347 |
msmarco_v2.1_doc_00_8128842#3_12552923
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http://2001-2009.state.gov/g/oes/continentalshelf/
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Defining the Limits of the U.S. Continental Shelf
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Defining the Limits of the U.S. Continental Shelf
Defining the Limits of the U.S. Continental Shelf
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Sovereign rights that apply to the EEZ, especially rights to the resources of the water column (e.g., pelagic fisheries) do not apply to the ECS. Establishing ECS limits will define the U.S. continental shelf in concrete geographical terms. Moreover, the United States has an inherent national interest in knowing, and declaring to others with specificity and certainty, the extent of sovereign rights with regard to the U.S. continental shelf beyond 200 nautical miles. Such certainty and international recognition is important to establishing the stability necessary for development and conservation of these potentially resource-rich areas. The collection and analysis of the data necessary to support the establishment of the U.S. ECS will, in itself, serve a range of other environmental, geologic, engineering, and resource management needs. The data will provide a better scientific understanding of formation and transformation processes of our continental margins. The United States will gain specific insights related to such areas as climate variability, marine ecosystems, undiscovered or unconventional energy, mineral resources, and hazards resulting from extreme events, such as earthquakes and tsunamis. Finally, exploration of little known areas, particularly in the ice-covered Arctic, will advance our operational capabilities and open new windows on this remote and inaccessible environment. Data Collection and Analysis
In late 2001, Congress directed the University of New Hampshire’s Joint Hydrographic Center (JHC) -- a partnership with NOAA -- to conduct a study that evaluated current data holdings relevant to establishing the U.S. ECS, and to recommend what additional data would be needed. This study identified a number of areas where the United States may have extended continental shelf:
| 2,500 | 4,319 |
msmarco_v2.1_doc_00_8128842#4_12555092
|
http://2001-2009.state.gov/g/oes/continentalshelf/
|
Defining the Limits of the U.S. Continental Shelf
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Defining the Limits of the U.S. Continental Shelf
Defining the Limits of the U.S. Continental Shelf
|
The data will provide a better scientific understanding of formation and transformation processes of our continental margins. The United States will gain specific insights related to such areas as climate variability, marine ecosystems, undiscovered or unconventional energy, mineral resources, and hazards resulting from extreme events, such as earthquakes and tsunamis. Finally, exploration of little known areas, particularly in the ice-covered Arctic, will advance our operational capabilities and open new windows on this remote and inaccessible environment. Data Collection and Analysis
In late 2001, Congress directed the University of New Hampshire’s Joint Hydrographic Center (JHC) -- a partnership with NOAA -- to conduct a study that evaluated current data holdings relevant to establishing the U.S. ECS, and to recommend what additional data would be needed. This study identified a number of areas where the United States may have extended continental shelf: the Atlantic East Coast, the Gulf of Mexico, the Gulf of Alaska, the Bering Sea, the Arctic Ocean, Kingman Reef/Palmyra Atoll, and the Mariana Islands/Guam. This amounts to about one million square kilometers or approximately twice the size of California. Roughly half of that area is likely to exist off Alaska. Additional analyses and data collection suggest an even larger ECS, in these and possibly other areas. As additional data are collected and existing data analyzed, we will begin to come to a more definitive conclusion as to the extent of the U.S. ECS.
| 3,348 | 4,884 |
msmarco_v2.1_doc_00_8128842#5_12556978
|
http://2001-2009.state.gov/g/oes/continentalshelf/
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Defining the Limits of the U.S. Continental Shelf
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Defining the Limits of the U.S. Continental Shelf
Defining the Limits of the U.S. Continental Shelf
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the Atlantic East Coast, the Gulf of Mexico, the Gulf of Alaska, the Bering Sea, the Arctic Ocean, Kingman Reef/Palmyra Atoll, and the Mariana Islands/Guam. This amounts to about one million square kilometers or approximately twice the size of California. Roughly half of that area is likely to exist off Alaska. Additional analyses and data collection suggest an even larger ECS, in these and possibly other areas. As additional data are collected and existing data analyzed, we will begin to come to a more definitive conclusion as to the extent of the U.S. ECS. Since 2002, the JHC has continued to receive grants from NOAA as directed by Congress to collect the bathymetric data specified in the study. The JHC has collected more than one million square kilometers of bathymetric data from eleven cruises: Arctic Ocean (2003, 2004, 2007), Gulf of Alaska (2005), Gulf of Mexico (2007), Atlantic Ocean (2004, 2005, 2008), Northern Mariana Islands and Guam (2006, 2007), and Bering Sea (2003). A cruise is planned for an area off Kingman Reef and Palmyra Atoll in 2008 or early 2009. All data collected thus far by the United States in support of defining its continental shelf have been released to the public.
| 4,320 | 5,532 |
msmarco_v2.1_doc_00_8128842#6_12558534
|
http://2001-2009.state.gov/g/oes/continentalshelf/
|
Defining the Limits of the U.S. Continental Shelf
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Defining the Limits of the U.S. Continental Shelf
Defining the Limits of the U.S. Continental Shelf
|
Since 2002, the JHC has continued to receive grants from NOAA as directed by Congress to collect the bathymetric data specified in the study. The JHC has collected more than one million square kilometers of bathymetric data from eleven cruises: Arctic Ocean (2003, 2004, 2007), Gulf of Alaska (2005), Gulf of Mexico (2007), Atlantic Ocean (2004, 2005, 2008), Northern Mariana Islands and Guam (2006, 2007), and Bering Sea (2003). A cruise is planned for an area off Kingman Reef and Palmyra Atoll in 2008 or early 2009. All data collected thus far by the United States in support of defining its continental shelf have been released to the public. The bathymetric data is available from the National Geophysical Data Center and the Joint Hydrographic Center. Useful Links
National Geophysical Data Center (NGDC)
Joint Hydrographic Center at the University of New Hampshire
U.S. Coast Guard Cutter Healy
For Additional Information
Matt Cassetta – Public Affairs Officer at the U.S. Department of State
| 4,884 | 5,885 |
msmarco_v2.1_doc_00_8135098#0_12559888
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http://2001-2009.state.gov/g/oes/id/index.htm
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Infectious Disease & Non-Infectious Disease
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Infectious Disease & Non-Infectious Disease
Infectious Disease & Non-Infectious Disease
Infectious Disease
Chronic Disease
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Infectious Disease & Non-Infectious Disease
Infectious Disease & Non-Infectious Disease
Infectious diseases, including HIV/AIDS, tuberculosis, malaria, polio, and several neglected tropical diseases (NTDs) are easily spread through personal contact, water, and air, (many NTDs are vector borne – transmitted by mosquitoes, flies, etc) and are a particularly significant problem in developing countries. In the past, infectious diseases have been widespread in developing countries and chronic diseases were found primarily in high income countries. However, the global pattern of disease burden is shifting. While infectious disease still remains a major problem in many countries, chronic diseases, including such noncommunicable conditions as cardiovascular disease, cancer, diabetes and respiratory disease, are now the major cause of death and disability, not only in developed countries, but also worldwide. The greatest total numbers of chronic disease deaths and illnesses now occur in developing countries. Infectious Disease
The U.S. commitment to combat infectious diseases has saved lives and reduced human suffering throughout the developing world. Through the President’s Emergency Plan for AIDS Relief (PEPFAR), for example, the U.S. supports lifesaving treatment for over 1.7 million people worldwide and has enabled 200,000 children to be born HIV-free. The 2008 reauthorization of PEPFAR expands the U.S. commitment to $48 billion over 5 years to combat HIV/AIDS, tuberculosis, and malaria. In addition to HIV/AIDS, tuberculosis (TB) remains a global public health problem and is one of the three leading causes of deaths worldwide due to infectious diseases. In light of this, the U.S. is on the frontline of the battle against tuberculosis (TB).
| 0 | 1,808 |
msmarco_v2.1_doc_00_8135098#1_12562110
|
http://2001-2009.state.gov/g/oes/id/index.htm
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Infectious Disease & Non-Infectious Disease
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Infectious Disease & Non-Infectious Disease
Infectious Disease & Non-Infectious Disease
Infectious Disease
Chronic Disease
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Infectious Disease
The U.S. commitment to combat infectious diseases has saved lives and reduced human suffering throughout the developing world. Through the President’s Emergency Plan for AIDS Relief (PEPFAR), for example, the U.S. supports lifesaving treatment for over 1.7 million people worldwide and has enabled 200,000 children to be born HIV-free. The 2008 reauthorization of PEPFAR expands the U.S. commitment to $48 billion over 5 years to combat HIV/AIDS, tuberculosis, and malaria. In addition to HIV/AIDS, tuberculosis (TB) remains a global public health problem and is one of the three leading causes of deaths worldwide due to infectious diseases. In light of this, the U.S. is on the frontline of the battle against tuberculosis (TB). In collaboration with host nation TB programs, the U.S. works to improve the quality of basic TB programs or DOTS (Directly Observed Therapy, Short Course) services; upgrade laboratory infrastructure; build a foundation to introduce new diagnostic technologies; and work with the World Health Organization (WHO) and other partners to conduct drug resistance surveys and surveillance. The U.S. Agency for International Development (USAID) is the lead USG agency in international TB control programs, with PEPFAR taking the lead role in TB/HIV co-infection, and the U.S. Department of Health and Human Services, Centers for Disease Control and Prevention (HHS/CDC) providing critical technical support to global and country level initiatives.
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