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.

Looking for Pythagorean pair-doubles

In teaching a geometry course, I find that I frequently need a Pythagorean triple. A Pythagorean triple, of course, is three whole numbers that satisfy the Pythagorean theorem to make a right triangle: 3, 4, 5, or 7, 40, 41.

But even more frequently, I find I need a Pythagorean pair-double: two sets of whole number-length legs with the same hypotenuse. For example, 5, 5 and 1, 7 are a pair-double, since both have the same hypotenuse. In this way, I can make an isosceles triangle that requires a little work to prove it isosceles.

Here's the first ten pair-doubles.