International Mathematical Olympiad 2004 Problem 5

In a convex quadrilateral $ABCD$ the diagonal $BD$ does not bisect the angles $\angle ABC$ and $\angle CDA$. The point $P$ lies inside $ABCD$ and satisfies $$\angle PBC = \angle DBA \text{ and } \angle PDC = \angle BDA.$$

Prove that $ABCD$ is a cyclic quadrilateral if and only if $AP = CP$.