NZ-NZMO 2022 Problem 7

Let $M$ be the midpoint of side $BC$ of acute triangle $ABC$ . The circle centered at $M$ passing through $A$ intersects the lines $AB$ and $AC$ again at $P$ and $Q$ , respectively. The tangents to this circle at $P$ and $Q$ meet at $D$ . Prove that the perpendicular bisector of $BC$ bisects segment $AD$ .