Question 89
Let $ABC$ be an acute triangle with $BC\gt AC$. The perpendicular bisector of side $AB$ meets the line $BC$ at $X$ and the line $AC$ at $Y$. Let $P$ be the orthogonal projection of $X$ onto $AC$, and let $Q$ be the orthogonal projection of $Y$ onto $BC$. Prove that the line $PQ$ meets the segment $AB$ at its midpoint.