Question 164
Let $AD$ be the median from $A$ of a triangle $ABC$. Let $M$ be a variable point on the given line $d$ perpendicular to $AD$ and denote by $E$ and $F$ respectively the midpoints of $MB$ and $MC$. The line through $E$ and perpendicular to $d$ meets $AB$ at $P$, the line through $F$ and perpendicular to $d$ meets $AC$ at $Q$. Prove that the line through $M$ and perpendicular to $PQ$ passes through a fixed point.