My answer to a question in Quora: Can you derive the Pythagorean theorem?
Yes, I can. You can, too, — if you learn a bit of geometry. A representative collection of proofs — very different, but every single of them is a valid and rigorous proof on its own – can be found here: Pythagorean Theorem.
When a theorem has so many proofs it is a sure sign that it holds a central place is some big chunk of mathematics — so many other bits are connected to it. Something similar happens with Euclid’s great theorem about the infinitude of the set of prime numbers — there are so many proofs for it, and the range of mathematics used is even wider than for the Pythagorean theorem. These proofs are really worth studying.