Team Selection Test for IMO

Let $ABC$ be a triangle with $m(\hat{B})>m(\hat{C})$. Interior and exterior angle bisectors at vertex $A$ intersect $BC$ at $D$ and $E$ respectively. A variable point $P$ lies on the ray $[EA$ such that $A$ is closer to $E$ than $P$. Lines $DP$ and $AC$ intersect at point $M$ and lines $ME$ and $AD$ intersect at point $Q$. Prove that all lines $PQ$ when a variable point $P$ changes intersect at a unique point. (Ali Adali).