Inscribed Squares in Right-Angled Triangle

Theorem
For any right-angled triangle, two squares can be inscribed inside it.

One square would share a vertex with the right-angled vertex of the right-angled triangle:
 * Inscribed-square-r.png

The other square would have a side lying on the hypotenuse of the right-angled triangle:
 * Inscribed-square-h.png

Side Lengths
Let $a, b, c$ be the side lengths of the right-angled triangle, where $c$ is the length of the hypotenuse.

Then the side lengths of the inscribed squares are given by:

Proof
By definition of inscribed polygon, all four vertices of the inscribed square lies on the sides of the right-angled triangle.