Inscribed Squares in Right-Angled Triangle/Compass and Straightedge Construction

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:

The other square would have a side lying on the hypotenuse of the right-angled triangle:

Both inscribed squares can be constructed using compass and straightedge, thus demonstrating their existence:

Shared right angle

Side lies on Hypotenuse