Problem

Source: 2020 Korea Junior Math Olympiad p4 KJMO

Tags: collinear, geometry



In an acute triangle $ABC$ with $\overline{AB} > \overline{AC}$, let $D, E, F$ be the feet of the altitudes from $A, B, C$, respectively. Let $P$ be an intersection of lines $EF$ and $BC$, and let $Q$ be a point on the segment $BD$ such that $\angle QFD = \angle EPC$. Let $O, H$ denote the circumcenter and the orthocenter of triangle $ABC$, respectively. Suppose that $OH$ is perpendicular to $AQ$. Prove that $P, O, H$ are collinear.