Question 34
Let $ABC$ be a scalene acute triangle $ABC$, $D$ be the orthogonal projection of $A$ on $BC$, $M$ and $N$ are the midpoints of $AB$ and $AC$ respectively. Let $P, Q$ are points on the minor arcs $\widehat{AB}$ and $\widehat{AC}$ of circumcircle of $\triangle ABC$ respectively such that $PQ \parallel BC$. Show that the circumcircles of $\triangle DPQ$ and $\triangle MND$ are tangent to each other if and only if $PQ$ passes through $M$.