Question 7
Let $ABC$ be a triangle with median $AD$. Let a line $d$ be perpendicular to $AD$, and let $M$ vary on $d$. Let $E,F$ be the midpoints of $MB,MC$, respectively. The line through $E$ perpendicular to $d$ meets $AB$ at $P$, and the line through $F$ perpendicular to $d$ meets $AC$ at $Q$. Prove that the line through $M$ perpendicular to $PQ$ always passes through a fixed point as $M$ varies on $d$.