## Maximum Size of Square Inscribed in Right Angled Triabgle of Given Base and Height

The diagram shows a square inscribed inside a right angled triangle.

The equation of the line from O to C is
$y=x$
anf the equation of the line BH is
$y= - \frac{h}{b} x + h$
. From these we can write
$c=- \frac{h}{b}c + h$
,br> Hence
$c(1+ \frac{h}{b})=h \rightarrow c= \frac{hb}{h+b}$
.
The square has side
$\frac{hb}{h+b}$
.