Question 39
Let $H$ be the orthocenter of triangle $ABC$ and $\Gamma$ its circumcircle. Let $P$ be a point on $\Gamma$ different from $A$, $B$, $C$, and let $M$ be the midpoint of segment $HP$. Choose points $D$, $E$, $F$ on lines $BC$, $CA$, $AB$, respectively, such that $AP \parallel HD$, $BP \parallel HE$, $CP \parallel HF$. Prove that $D$, $E$, $F$, $M$ are collinear.