Jigsaw-style proofs of the Pythagorean theorem with non-square squares The two squares on the legs of a right triangle can be chopped up (or "dissected") into several pieces that can be reassembled jigsaw-style into a square congruent to that whose side is the hypotenuse. If a plane region of some other shape than a square is used, with a side having the length of one of the sides of the triangle, the theorem remains true if the same shape is glued onto all three sides. If a different shape is used, might this dissection proof become simpler or more comprehensible or more enlightening or otherwise better? If not, can that negative result be made precise and proved?

The following is a minimalist answer. Let $\triangle ABC$ be right-angled at $C$. Drop a perpendicular from $C$ to $P$ on $AB$. This divides $\triangle ABC$ into two similar right triangles which, without any cutting at all, can be reassembled (without any motion) to make the triangle on the hypotenuse.

If we prefer the figures to be erected "outside" the original triangle, reflect the pieces across the sides.