Question 117
Let $ABC$ be a scalene acute triangle with incenter $I$ and circumcircle $\Gamma$. The line $AI$ meets $\Gamma$ again at the point $M$. Let $N$ be the midpoint of $BC$, and let $T$ be the point on $\Gamma$ such that $IN \perp MT$. Let $\ell$ be the line through $I$ perpendicular to $AI$. Let $\ell$ meet the lines $TB$ and $TC$ at the points $P$ and $Q$ respectively. Prove that $PB = CQ$.