## 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

\[ \frac{hb}{h+b}\]

.