Proving the Pythagorean theorem with similarity

Here is a right triangle, split along the altitude to the hypotenuse.

This creates three similar triangles: ABD, CAD, and CBA. Using similarity, you can show

$$a^2=cx$$
and also
$$b^2=c^2-cx$$
which is all you need to show that
$$a^2+b^2=c^2$$
The neat part is that it requires no argument about areas at all.