Problem

Source: JMO/5 2016

Tags: geometry, similar triangles, 2016 USAJMO, USAJMO, USA(J)MO, isogonal conjugates, Angle Chasing



Let $\triangle ABC$ be an acute triangle, with $O$ as its circumcenter. Point $H$ is the foot of the perpendicular from $A$ to line $\overleftrightarrow{BC}$, and points $P$ and $Q$ are the feet of the perpendiculars from $H$ to the lines $\overleftrightarrow{AB}$ and $\overleftrightarrow{AC}$, respectively. Given that $$AH^2=2\cdot AO^2,$$prove that the points $O,P,$ and $Q$ are collinear.