Proving the Pythagorean Theorem

Here is a new paper with complicated proofs of the Pythagorean theorem. I don't know why anyone bothers, as plenty of proofs are very simple. Some on the Wikipedia page do not even require any words.

My favorite depends on the fact that in similar figures, all linear dimensions are proportional. Areas are proportional to the square of the linear dimensions.

Given a right triangle, drop a perpendicular from the right angle to the hypotenuse. This makes two new triangles, both similar to the original, and combining to make up the original.

Since the triangles are similar, the areas are proportional to the square of their hypotenuses.

The three hypotenuses are the three sides of the original triangle, and adding the areas of the new triangles gives the area of the original, so adding the squares of the sides must give the area of the hypotenuse.

I like this proof because it is so simple and direct. It does not rely on any tricky cancelations.