Circumcircle intersecting circumcircle

Question

The tangent to the circumcircle of $\triangle ABC$ at $B$ and $C$ meet at $D$. Circumcircle of $\triangle BCD$ meet $AC$ and $AB$ again at $E$ and $F$ respectively. Let $O$ be the circumcenter of $\triangle ABC$. Show that $AO$ is perpendicular to $EF$.

This question was sent to me by a friend who loves math. I took many attempts to solve this. I tried to work from the result to get the given but couldn't solve it. Any help would be appreciated. Thanks in advance.

Answer 1

Let $\angle ACB = x$ and let line $AO$ meet line $EF$ at $G$. Then

• $\angle BFE = \angle BEC = \angle BCA = x$
• Since $\angle AOB = 2\angle BCA = 2x$ we have $ \angle BAO (= \angle ABO) = 90-x$
• Observe triangle $AFG$...