Pythagorean Theorem

The Pythagorean Theorem, sometimes known as Pythagoras' Theorem, is a famous relationship among the legs, $$a$$ and $$b$$, and the hypotenuse, $$c$$, of a right triangle. For any such triangle, the equality $$a^2 + b^2 = c^2$$ holds.

Pythagoras' Theorem has a wide array of uses in mathematics, usually applied as a crucial step in many geometrical problems. Dozens of proofs have been developed to show this identity, many of which are featured in Wikipedia's article.

Pythagorean Triples
Occasionally, the sum of two integer squares equals a third integer square. For example, $$3^2 + 4^2 = 5^2$$. In general,
 * $$a = m^2 - n^2,\; b= 2mn,\; c = m^2 + n^2$$

is a quick way to generate Pythagorean Triples. Obviously, integer multiples of Pythagorean Triples are also Pythagorean Triples, thus justifying the claim that there are infinitely many such triples.

Common Pythagorean Triples
There are infinitely many phythagorean triples but there are several triples that are worthwhile to memorize.


 * $$ 3^2 + 4^2 = 5^2$$
 * $$ 5^2 + 12^2 = 13^2$$
 * $$ 7^2 + 24^2 = 25^2$$
 * $$ 9^2 + 40^2 = 41^2$$

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