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For example, (3, 4, 5) is a primitive Pythagorean triple whereas (6, 8, 10) is not. Every Pythagorean triple can be scaled to a unique primitive Pythagorean triple by dividing (a, b, c) by their greatest common divisor. Conversely, every Pythagorean triple can be obtained by multiplying the elements of a primitive Pythagorean triple by a ...
[9] [10] The Tetractys [also known as the decad] is an equilateral triangle formed from the sequence of the first ten numbers aligned in four rows. It is both a mathematical idea and a metaphysical symbol that embraces within itself—in seedlike form—the principles of the natural world, the harmony of the cosmos, the ascent to the divine ...
With a the shorter and b the longer legs of a triangle and c its hypotenuse, the Pythagoras family of triplets is defined by c − b = 1, the Plato family by c − b = 2, and the Fermat family by | a − b | = 1. The Stifel sequence produces all primitive triplets of the Pythagoras family, and the Ozanam sequence produces all primitive triples ...
[6]: p.7 For example, parent (3, 4, 5) has excircle radii equal to 2, 3 and 6. These are precisely the inradii of the three children (5, 12, 13), (15, 8, 17) and (21, 20, 29) respectively. If either of A or C is applied repeatedly from any Pythagorean triple used as an initial condition, then the dynamics of any of a , b , and c can be ...
If a right triangle has integer side lengths a, b, c (necessarily satisfying the Pythagorean theorem a 2 + b 2 = c 2), then (a,b,c) is known as a Pythagorean triple. As Martin (1875) describes, the Pell numbers can be used to form Pythagorean triples in which a and b are one unit apart, corresponding to right triangles that are nearly isosceles.
The Pythagorean triple (4,3,5) is associated to the rational point (4/5,3/5) on the unit circle. In mathematics, the rational points on the unit circle are those points (x, y) such that both x and y are rational numbers ("fractions") and satisfy x 2 + y 2 = 1. The set of such points turns out to be closely related to primitive Pythagorean triples.
The Pythagorean theorem was known and used by the Babylonians and Indians centuries before Pythagoras, [216] [214] [217] [218] but he may have been the first to introduce it to the Greeks. [ 219 ] [ 217 ] Some historians of mathematics have even suggested that he—or his students—may have constructed the first proof . [ 220 ]
An example of a ternary relation in elementary geometry can be given on triples of points, where a triple is in the relation if the three points are collinear. Another geometric example can be obtained by considering triples consisting of two points and a line, where a triple is in the ternary relation if the two points determine (are incident ...