Question 1
In an acute triangle $ABC$, $AC>AB$, $D$ is the point on $BC$ such that $AD=AB$. Let $\omega_1$ be the circle through $C$ tangent to $AD$ at $D$, and $\omega_2$ the circle through $C$ tangent to $AB$ at $B$. Let $F\ (\ne C)$ be the second intersection of $\omega_1$ and $\omega_2$. Prove that $F$ lies on $AC$.