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The (3, 4, 5) triangle and its multiples are the only integer right triangles having sides in arithmetic progression. [39] The (4, 5, 6) triangle and its multiples are the only triangles with one angle being twice another and having integer sides in arithmetic progression. [39]
For instance, the triangle with sides = = and = is a right triangle, but (,,) is not a Pythagorean triple because the square root of 2 is not an integer or ratio of integers. Moreover, 1 {\displaystyle 1} and 2 {\displaystyle {\sqrt {2}}} do not have an integer common multiple because 2 {\displaystyle {\sqrt {2}}} is irrational .
However, infinitely many almost-isosceles right triangles do exist. These are right-angled triangles with integer sides for which the lengths of the non-hypotenuse edges differ by one. [5] [6] Such almost-isosceles right-angled triangles can be obtained recursively, a 0 = 1, b 0 = 2 a n = 2b n−1 + a n−1 b n = 2a n + b n−1. a n is length ...
In other words, a Pythagorean triple represents the lengths of the sides of a right triangle where all three sides have integer lengths. [1] Such a triple is commonly written (a, b, c). Some well-known examples are (3, 4, 5) and (5, 12, 13). A primitive Pythagorean triple is one in which a, b and c are coprime (the greatest common divisor of a ...
There is a method to construct all Pythagorean triples that contain a given positive integer x as one of the legs of the right-angled triangle associated with the triple. It means finding all right triangles whose sides have integer measures, with one leg predetermined as a given cathetus. [13] The formulas read as follows.
For example, the Heronian triangle of side lengths ,, and area 72, since none of its altitudes is an integer. Such Heronian triangles are known as indecomposable. [6] However, every Heronian triangle can be constructed from right triangles with rational side lengths, and is thus similar to a decomposable Heronian triangle. In fact, at least one ...
Equivalently, by the Pythagorean theorem, they are the odd prime numbers for which is the length of the hypotenuse of a right triangle with integer legs, and they are also the prime numbers for which itself is the hypotenuse of a primitive Pythagorean triangle. For instance, the number 5 is a Pythagorean prime; is the hypotenuse of a right ...
More generally, the problem of finding all equable triangles with integer sides (that is, equable Heronian triangles) was considered by B. Yates in 1858. [ 5 ] [ 6 ] As W. A. Whitworth and D. Biddle proved in 1904, there are exactly three solutions, beyond the right triangles already listed, with sides (6,25,29), (7,15,20), and (9,10,17).