There is a large class of proofs of the Pythagorean Theorem which show that a square of side length $c$ can be dissected into squares of side lengths $a$ and $b$.
There is also the proof which shows that the right triangle itself can be dissected into similar triangles whose hypotenuses have length $a$ and $b$.
Are there any natural-looking proofs which dissect some shape other than a square or right triangle, like a dissection of an equilateral triangle of side length $c$ into equilateral triangles of side lengths $a$ and $b$? (By Bolyai's theorem we know that such a dissection exists, but it might be really painful to construct...)