AT-OEMO-FED1 2009 Problem 4

Let $D, E$ , and $F$ be respectively the midpoints of the sides $BC, CA$ , and $AB$ of $\vartriangle ABC$ . Let $H_a, H_b, H_c$ be the feet of perpendiculars from $A, B, C$ to the opposite sides, respectively. Let $P, Q, R$ be the midpoints of the $H_bH_c, H_cH_a$ , and $H_aH_b$ respectively. Prove that $PD, QE$ , and $RF$ are concurrent.