MathNet 2013 Question 76

Let points $A$ and $B$ be on circle $\omega$ centered at $O$. Suppose that $\omega_{A}$ and $\omega_{B}$ are circles not containing $O$ which are internally tangent to $\omega$ at $A$ and $B$, respectively. Let $\omega_{A}$ and $\omega_{B}$ intersect at $C$ and $D$ such that $D$ is inside triangle $A B C$. Suppose that line $B C$ meets $\omega$ again at $E$ and let line $E A$ intersect $\omega_{A}$ at $F$. If $F C \perp C D$, prove that $O, C$, and $D$ are collinear.

Supplemental illustration (AI-generated):