Question 80
A triangle $ABC$ is given. Call its orthocenter $H$. Define $\omega$ as the circle through $B$, $C$, and $H$, and define $\Gamma$ as the circle with diameter $AH$. Let $X$ be the other intersection of $\omega$ and $\Gamma$, and let the reflection of $\Gamma$ over $AX$ be $\gamma$.
Suppose $\gamma$ and $\omega$ intersect again at $Y \neq X$, and line $AH$ and $\omega$ intersect again at $Z \neq A$. Show that the circle through $A$, $Y$, $Z$ passes through the midpoint of segment $BC$.
Suppose $\gamma$ and $\omega$ intersect again at $Y \neq X$, and line $AH$ and $\omega$ intersect again at $Z \neq A$. Show that the circle through $A$, $Y$, $Z$ passes through the midpoint of segment $BC$.