The Pythagorean Theorem is one of the most interesting theorems for two reasons: First, it’s very elementary; even high school students know it by heart. Second, it has hundreds of proofs. The proof below uses triangle similarity.

**Pythagorean Theorem**

In a right triangle with side lengths and and hypotenuse , the following equation always holds:

.

**Proof**

In the figure above, triangle is similar to triangle . In similar triangles, the corresponding sides are proportional. It follows that

which is equivalent to

.

Cross multiplying, we have . Adding to both sides of the equation, we have , which is what we want to prove.

Nice proof! I’ve seen a version of this one where the hypotenuse of the right triangle is the diameter of the circle. I don’t think I’ve seen the idea of using the hypotenuse as the radius like this.

It’s actually a derivative of one of the proofs I’ve seen. I’m searching for the same proof on the net. It could be an addition to the hundreds of proofs.