Pythagorean Theorem
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The Pythagorean Theorem states that for all right triangles, 
, where c is the hypotenuse, and a and b are the legs of the right triangle. This theorem is a classic to prove, and hundreds of proofs have been published. Many proofs of it don't have any words and are simply pictures that challenge the reader to figure out how the picture works. The Pythagorean Theorem is one of the most frequently used theorem in geometry, and is one of the many tools in a good geometer's arsenal. A very large number of geometry problems can be solved by building right triangles and applying the Pythagorean Theorem.
This is generalized by the Pythagorean Inequality (See geometric inequalities) and the Law of Cosines.)
Introductory
Example Problems
Common Pythagorean Triples
A Pythagorean Triple is a set of 3 positive integers such that 
, i.e. the 3 numbers can be the lengths of the sides of a right triangle. Among these, the Primitive Pythagorean Triples, those in which the three numbers have no common divisor, are most interesting. A few of them are:
3-4-5
5-12-13
7-24-25
8-15-17
9-40-41
