Question 135
Given a non-isosceles acute-angled triangle $\triangle ABC$ where $O$ is the midpoint of $BC$. Let the circle with diameter $BC$ intersect $AB$, $AC$ at $D$, $E$ respectively. Let the angle bisectors of $\angle A$ and $\angle DOE$ intersect at $P$. If the circumcircles of $\triangle BPD$ and $\triangle CPE$ intersect at $P$ and $Q$, show that $Q$ lies on $BC$.