Iranian Mathematical Olympiad

Consider a triangle $ABC$ with incircle $\omega$ that is respectively tangent to sides $BC$, $CA$ and $AB$ at $D$, $E$ and $F$. Points $P$, $Q$ are inside of $\angle A$ so that $FP=FB$, $FP\parallel AC$ and $EQ=EC$, $EQ\parallel AB$. Prove that $P$, $Q$ and $D$ are collinear.