Question 9
Let $ABC$ be an acute-angled triangle with $AB \lt AC$, incenter $I$, and let $M$ be the midpoint of major arc $BAC$. Suppose the perpendicular line from $A$ to segment $BC$ meets lines $BI$, $CI$, and $MI$ at points $P$, $Q$, and $K$ respectively. Prove that the $A$-median line in $\triangle AIK$ passes through the circumcentre of $\triangle PIQ$.