Inscribed Squares in Right-Angled Triangle/Side Lengths

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:

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 $l$ of the inscribed squares are given by:

Shared right angle

$l = \dfrac {a b} {a + b}$

Side lies on Hypotenuse

$l = \dfrac {a b c} {a b + c^2}$