MathNet 2016-part-06 — 184 problems (EN–VI) Question 117

Let $ABC$ be a triangle with $AB \neq AC$. Let the angle bisector of $\angle BAC$ intersect $BC$ at $P$, and intersect the perpendicular bisector of segment $BC$ at $Q$. Prove that $\frac{PQ}{AQ} = \left(\frac{BC}{AB+AC}\right)^2$.