Question 5
Let $ABC$ be an acute, non-isosceles triangle with circumcenter $O$ and incenter $I$. The incircle $(I)$ touches $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. Suppose that $EF$ meets $(O)$ at $P$ and $Q$. Prove that the circle $(PQD)$ passes through the midpoint of $BC$.