Question 33
Let $\triangle ABC$ be an acute triangle with pairwise distinct side lengths, and let $O$, $H$ be its circumcenter and orthocenter, respectively.
The line $OA$ meets the altitude of $\triangle ABC$ through $B$ and the altitude through $C$ at the points $P$, $Q$, respectively.
Prove that the circumcenter of triangle $PQH$ lies on one of the medians of triangle $ABC$.
(Note: in a triangle, the segment joining a vertex to the midpoint of the opposite side is called a median.)
The line $OA$ meets the altitude of $\triangle ABC$ through $B$ and the altitude through $C$ at the points $P$, $Q$, respectively.
Prove that the circumcenter of triangle $PQH$ lies on one of the medians of triangle $ABC$.
(Note: in a triangle, the segment joining a vertex to the midpoint of the opposite side is called a median.)