Question 165
Let $ABC$ be a triangle with $\angle A = 60^\circ$, orthocenter $H$ and centroid $G$. Let $A'$ be the point on the circumcircle of $\triangle ABC$ diametrically opposite to $A$. Let $M$ be the midpoint of $BC$. The line through $M$ parallel to $AA'$ intersects $BH, CH$ at $K, L$, respectively. $P$ and $Q$ are distinct points on $GA'$ such that $GP = GQ = \frac{1}{2}GH$. Let $D$ be the reflection of $A$ over $GA'$. Prove that the circumcircle of $\triangle DPQ$ is tangent to the circumcircle of $\triangle HKL$.