OIM 2015 Problem 2

A line $r$ contains the points $A$ , $B$ , $C$ , $D$ in that order. Let $P$ be a point not in $r$ such that $\angle{APB} = \angle{CPD}$ . Prove that the angle bisector of $\angle{APD}$ intersects the line $r$ at a point $G$ such that: $\frac{1}{GA} + \frac{1}{GC} = \frac{1}{GB} + \frac{1}{GD}$