TH-TMO 2016 Problem 1

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