OMCC 2019 Problem 3

Let $ABC$ be a triangle and $\Gamma$ its circumcircle. Let $D$ be the foot of the altitude from $A$ to the side $BC$ , $M$ and $N$ the midpoints of $AB$ and $AC$ , and $Q$ the point on $\Gamma$ diametrically opposite to $A$ . Let $E$ be the midpoint of $DQ$ . Show that the lines perpendicular to $EM$ and $EN$ passing through $M$ and $N$ , respectively, meet on $AD$ .