Inscribed Squares in Right-Angled Triangle/Side Lengths/Shared Right Angle

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

Then the side length $l$ of the inscribed square that shares a right angle with the right-angled triangle is given by:
 * $l = \dfrac {a b} {a + b}$

Proof

 * Inscribed-square-r.png

In the figure above, let $BC = a$ and $AC = b$.

Note that $DE \parallel CF$.

Therefore $\triangle BDE \sim \triangle BCA$ by Equiangular Triangles are Similar.

Thus:

Also see

 * Haidao Suanjing/Examples/Example 1