Of the hundreds upon hundreds of the known proofs of the Pythagorean theorem, Euclid’s proof has to be the most famous one. It is Proposition 47 of Book 1 of his immortal work, Elements. No copies of the original text survive, but all the known Greek versions and translations base the theorem’s proof on the same device. In mathematical folklore it is known variously as “bride’s chair”, “Franciscan’s cowl”, “peacock’s tail”, “windmill” or (in Russia) “Pythagorean pants”.
Probably the easiest way to understand the proof is to read it in the Oliver Byrne’s 1847 edition. Byrne aimed to present Euclid’s proofs in terms of pictures, using as little text — and in particular as few labels — as possible. Also, he went totally bonkers with color. I love it!
If you think the proof is unobvious and unnecessary complicated but strangely beautiful, you should know: Schopenhauer has described it as a “brilliant piece of perversity”.