Question 76
Let $H$ be the orthocenter of triangle $ABC$. Let the lines $BH$ and $AC$ meet at $E$, and let the lines $CH$ and $AB$ meet at $F$. Let $X$ be an arbitrary point on line $BC$. Let the circumcircle of triangle $BEX$ meet line $AB$ again at $Y$, and let the circumcircle of triangle $CFX$ meet line $AC$ again at $Z$. Prove that the circumcircle of triangle $AYZ$ is tangent to line $AH$.