Question

A circle is inscribed in an equilateral triangle of side $a$.The area of any square inscribed in this circle is

Answer 1

Let $a$ be the length of each side of the equilateral $△ABC,$
Then $r$ the radius of the incircle
$=\left(\frac{1}{3}\right)AP$
$\Rightarrow r=\frac{1}{3}\sqrt{a^2-\frac{a^2}{4}}=\frac{a}{2\sqrt{3}}$ on simplification.
Area of the square $PQRS$ inscribed in this circle
$={PQ}^2={OP}^2+{OQ}^2$
$\Rightarrow 2r^2=2\times \frac{a^2}{4\times 3}$
$\Rightarrow 2r^2=\frac{a^2}{6}$