CN-TST 2015 Problem 13

$\triangle{ABC}$ is isosceles with $AB = AC >BC$ . Let $D$ be a point in its interior such that $DA = DB+DC$ . Suppose that the perpendicular bisector of $AB$ meets the external angle bisector of $\angle{ADB}$ at $P$ , and let $Q$ be the intersection of the perpendicular bisector of $AC$ and the external angle bisector of $\angle{ADC}$ . Prove that $B,C,P,Q$ are concyclic.