Let $A B C$ be a triangle with orthocentre $H$, and let $D, E$, and $F$ denote the respective midpoints of line segments $A B, A C$, and $A H$. The reflections of $B$ and $C$ in $F$ are $P$ and $Q$, respectively. (a) Show that lines $P E$ and $Q D$ intersect on the circumcircle of triangle $A B C$. (b) Prove that lines $P D$ and $Q E$ intersect on line segment $A H$. (Merlijn Staps, the Netherlands)