IMAS 2015/2016 Junior Round 2 Question 15

$P$ and $Q$ are points on the bisector of the exterior angle at $A$ of triangle $ABC$, with $A$ between $P$ and $Q$, such that $BP$ is parallel to $CQ$. $D$ is the point on $BC$ such that $DP=DQ$. Prove that $AB$ is parallel to $DQ$.

Detailed solutions are needed.