Question 40
Let $\triangle A B C$ be a triangle with orthocentre $H$ and circumcircle $\Gamma$. Further, let $D$ be the reflection of $A$ in $B$, and let $E$ be the reflection of $A$ in $C$. We call the midpoint of segment $D E$ $M$.
Prove that the tangent to $\Gamma$ at $A$ is perpendicular to $H M$.
Prove that the tangent to $\Gamma$ at $A$ is perpendicular to $H M$.