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:

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

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:

$\ds \frac {CG} {CD}$

$=$

$\ds \frac {FH} {AB}$

$\ds \frac {\frac {a b} c - l} {\frac {a b} c}$

$=$

$\ds \frac l c$

$\ds \frac {a b - c l} {a b}$

$=$

$\ds \frac l c$

$\ds a b c - c^2 l$

$=$

$\ds a b l$

$\ds a b c$

$=$

$\ds a b l + c^2 l$

$\ds l$

$=$

$\ds \frac {a b c} {a b + c^2}$

$\blacksquare$