Question 19
Let $A B C$ be an acute-angled triangle with circumcircle $\omega$ and circumcentre $O$. Points $D \neq B$ and $E \neq C$ lie on $\omega$ such that $B D \perp A C$ and $C E \perp A B$. Let $C O$ meet $A B$ at $X$, and $B O$ meet $A C$ at $Y$.
Prove that the circumcircles of triangles $B X D$ and $C Y E$ have an intersection on line $A O$.
Prove that the circumcircles of triangles $B X D$ and $C Y E$ have an intersection on line $A O$.