Question 86
Let $ABC$ be an acute-angled triangle such that $|BC| \gt |AC|$. The perpendicular bisector of the segment $\overline{AB}$ intersects the side $BC$ at point $P$, and the line $AC$ at point $Q$. Let $R$ be the foot of the perpendicular from point $P$ to the side $\overline{AC}$, and let $S$ be the foot of the perpendicular from point $Q$ to the line $BC$. Prove that the line $RS$ bisects the segment $\overline{AB}$.