# Pythagorean Theorem

The **Pythagorean Theorem** states that for a right triangle with legs of length and and hypotenuse of length we have the relationship . This theorem has been know since antiquity and is a classic to prove; hundreds of proofs have been published and many can be demonstrated entirely visually(the book *The Pythagorean Proposition* alone consists of more than 370). The Pythagorean Theorem is one of the most frequently used theorems 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 and the Law of Cosines.

## Contents

## Proofs

In these proofs, we will let be any right triangle with a right angle at .

### Proof 1

We use to denote the area of triangle .

Let be the perpendicular to side from .

Since are similar right triangles, and the areas of similar triangles are proportionate to the squares of corresponding side lengths,

.

But since triangle is composed of triangles and , , so . ∎

### Proof 2

Consider a circle with center and radius . Since and are perpendicular, is tangent to . Let the line meet at and , as shown in the diagram:

Evidently, and . By considering the power of point with respect to , we see

. ∎

## 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: