Prove the Pythagorean theorem

Let $△ABC$ have its right angle at $C$. Write $a=BC$, $b=CA$, and $c=AB$.

Prove that
$$a^2+b^2=c^2.$$

Something over four hundred proofs are on record, and that abundance is itself the interesting fact. The theorem sits at a junction where area, similarity, algebra and trigonometry all meet, and each of them reaches it by a different road.

So post the proof you find most convincing, not merely the shortest — and say what it costs you. A proof that slides areas around is assuming something different from one that leans on similar triangles, and both are quietly standing on the parallel postulate. Naming the assumption is part of the answer.