$ABCD$ is a square of side $18$ cm. $F$ is a point inside the square, such that $BCF$ forms an equilateral triangle. $CFA$ is a quarter circle with centre $B$. $E$ is the point on $AB$ such that the area of the region $EFC$ is half that of the area of the quarter circle.
Find the length of $AE$ (in cm).
Express the area of $Area(FEBC)$ in two different ways, namely, $Area\left(FEC\right)+ Area\left(EBC\right)$ and $Area\left(FEB\right)+Area\left(FBC\right)$. Now equate them to get the value of $EB$ and thereafter $AE$.