Question 45
In acute triangle $A B C$, points $D$ and $E$ are the feet of the perpendiculars from $A$ to $B C$ and $B$ to $C A$, respectively. Segment $A D$ is a diameter of circle $\omega$. Circle $\omega$ intersects sides $A C$ and $A B$ at $F$ and $G$ (other than $A$), respectively. Segment $B E$ intersects segments $G D$ and $G F$ at $X$ and $Y$ respectively. Ray $D Y$ intersects side $A B$ at $Z$. Prove that lines $X Z$ and $B C$ are perpendicular.