DE-BUNDESWETTBEWERB 2016 Problem 3

Let $A,B,C$ and $D$ be points on a circle in this order. The chords $AC$ and $BD$ intersect in point $P$ . The perpendicular to $AC$ through C and the perpendicular to $BD$ through $D$ intersect in point $Q$ . Prove that the lines $AB$ and $PQ$ are perpendicular.