Question 74
Suppose that $O$ and $I$ are the centres of the circumcircle and incircle of $\triangle ABC$ with radius $R$ and $r$, respectively. $P$ is the midpoint of arc $BAC$. Let $QP$ be the diameter of $O$. Let $PI$ intersect $BC$ at point $D$, and let the circumcircle of $\triangle AID$ intersect the extended line of $PA$ at point $F$. Let point $E$ be on $PD$ such that $DE = DQ$. Prove that, if $\angle AEF = \angle APE$, then $\sin^2 \angle BAC = \frac{2r}{R}$.
(posed by Xiong Bin)
(posed by Xiong Bin)