USAJMO 2016 Question 5

Let $ABC$ be acute with circumcenter $O$. Let $H$ be the foot from $A$ to $BC$, and $P,Q$ the feet from $H$ to $AB,AC$. Given $AH^2=2AO^2$, prove that $O,P,Q$ are collinear.

Proposed by Zuming Feng, Jacek Fabrykowski.