ZA-SAMO 2008 Problem 5

Triangle $ABC$ has orthocentre $H$ . The feet of the perpendiculars from $H$ to the internal and external bisectors of $\hat{A}$ are $P$ and $Q$ respectively. Prove that $P$ is on the line that passes through $Q$ and the midpoint of $BC$ . (Note: The ortohcentre of a triangle is the point where the three altitudes intersect.)