BALTIC-WAY 2009 Problem 12

In a quadrilateral $ABCD$ we have $AB||CD$ and $AB=2CD$ . A line $\ell$ is perpendicular to $CD$ and contains the point $C$ . The circle with centre $D$ and radius $DA$ intersects the line $\ell$ at points $P$ and $Q$ . Prove that $AP\perp BQ$ .