Problem

Source: Serbia JBMO TST 2021

Tags: geometry, orthocenter, parallelogram, Circumcenter, circumcircle



On sides $AB$ and $AC$ of an acute triangle $\Delta ABC$, with orthocenter $H$ and circumcenter $O$, are given points $P$ and $Q$ respectively such that $APHQ$ is a parallelogram. Prove the following equality: \begin{align*} \frac{PB\cdot PQ}{QA\cdot QO}=2 \end{align*}