Inscribed Squares in Right-Angled Triangle/Side Lengths/Side Lies on Hypotenuse

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 has a side lying on the hypotenuse of the right-angled triangle is given by:
 * $l = \dfrac {a b c} {a b + c^2}$

Proof

 * Inscribed-square-h.png

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

$CD$ is drawn such that $AB \perp CD$.

Since $CD$ is the height of $\triangle ABC$:
 * $CD = \dfrac {a b} c$

Note that $FH \parallel AB$.

Therefore $\triangle CFH \sim \triangle CAB$ by Equiangular Triangles are Similar.

Thus: