MathNet 2021 — Part 01 Question 93

Let $I$ and $O$ be the incenter and the circumcenter of a triangle $ABC$, respectively, and let $s_a$ be the exterior bisector of angle $\angle BAC$. The line through $I$ perpendicular to $IO$ meets the lines $BC$ and $s_a$ at points $P$ and $Q$, respectively. Prove that $IQ = 2IP$.

**Proposed by Serbia**