Question 31
Let $\triangle ABC$ be a triangle with orthocentre $H$ and circumcircle $\Gamma$. Let $D$ be the reflection of $A$ across the point $B$, and let $E$ be the reflection of $A$ across the point $C$. Let $M$ be the midpoint of segment $DE$.
Prove that the tangent to $\Gamma$ at $A$ is perpendicular to $HM$.
Prove that the tangent to $\Gamma$ at $A$ is perpendicular to $HM$.