The **Pythagorean theorem** posits that in any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of both legs. This is perhaps one of the most oft-proven theorems. *The Pythagorean Proposition*, a book published in 1940, contains 370 proofs of Pythagoras' theorem, including one by American President James Garfield. Making the Pythagorean theorem useful for finding distance in 2-coordinate graphs and diagonals of rectangles.

## Theorem

Let be a triangle.

If is a right triangle with the right angle at , then

Conversely, if , then is a right triangle with the right angle at .

This is more commonly stated mathematically:

, assuming that is the length of the side opposite the right angle (the hypotenuse).