DE-MO 2017 Problem 4

Let $ABCD$ be a cyclic quadrilateral. The point $P$ is chosen on the line $AB$ such that the circle passing through $C,D$ and $P$ touches the line $AB$ . Similarly, the point $Q$ is chosen on the line $CD$ such that the circle passing through $A,B$ and $Q$ touches the line $CD$ .

Prove that the distance between $P$ and the line $CD$ equals the distance between $Q$ and $AB$ .