Question 99
Let $ABC$ be a triangle and let $I$ be the centre of the inscribed circle of this triangle. The line through $I$ perpendicular to $AI$ intersects the circumcircle of $\triangle ABC$ at the points $P$ and $Q$, where $P$ lies on the same side of $AI$ as $B$. Let $S$ be the second intersection point of the circumcircles of $\triangle BIP$ and $\triangle CIQ$. Prove that $SI$ is the bisector of $\angle PSQ$.
