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An integer triangle or integral triangle is a triangle all of whose side lengths are integers. A rational triangle is one whose side lengths are rational numbers ; any rational triangle can be rescaled by the lowest common denominator of the sides to obtain a similar integer triangle, so there is a close relationship between integer triangles ...
A Heronian triangle is commonly defined as one with integer sides whose area is also an integer. The lengths of the sides of such a triangle form a Heronian triple ( a, b, c ) for a ≤ b ≤ c . Every Pythagorean triple is a Heronian triple, because at least one of the legs a , b must be even in a Pythagorean triple, so the area ab /2 is an ...
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number , other examples being square numbers and cube numbers . The n th triangular number is the number of dots in the triangular arrangement with n dots on each side, and is equal to the sum of the n natural ...
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A Brahmagupta triangle is a triangle whose side lengths are consecutive positive integers and area is a positive integer. [1] [2] [3] The triangle whose side lengths are 3, 4, 5 is a Brahmagupta triangle and so also is the triangle whose side lengths are 13, 14, 15.
All of the right-angled triangles are similar, i.e. the ratios between their corresponding sides are the same. For sin, cos and tan the unit-length radius forms the hypotenuse of the triangle that defines them. The reciprocal identities arise as ratios of sides in the triangles where this unit line is no longer the hypotenuse.
Triangles based on Pythagorean triples are Heronian, meaning they have integer area as well as integer sides. The possible use of the 3 : 4 : 5 triangle in Ancient Egypt, with the supposed use of a knotted rope to lay out such a triangle, and the question whether Pythagoras' theorem was known at that time, have been much debated. [3]
The area of a rational-sided right triangle is called a congruent number, so no congruent number can be square. A right triangle and a square with equal areas cannot have all sides commensurate with each other. There do not exist two integer-sided right triangles in which the two legs of one triangle are the leg and hypotenuse of the other ...