in a sense, Euclid's proof of the P theorem, also involves the idea of moving congruent triangles:



his argument is to observe that triangles FBC and ABD are congruent, and that each is half of the most nearby pink area.

so, intuitively, the idea is to squeeze the upper pink squares into a parallelogram and then squeeze it into the pink rectangle.

it does not look as elegant as ZhouPei's proof though.