Question 14
Let $ABC$ be a triangle. Let $H_A$ be the foot of the altitude from $A$, and let $A'$ be the midpoint of $BC$. Let $Q_A$ be the reflection of $H_A$ about $A'$. Define $Q_B,Q_C$ analogously. Finally, let $R$ be the intersection other than $Q_A$ of the circumcircles of $Q_AQ_BC$ and $Q_ABQ_C$.
Prove that $Q_AR$ is perpendicular to $BC$.
Prove that $Q_AR$ is perpendicular to $BC$.