Question 99
Let $ABC$ be a triangle with $AC \gt AB \gt BC$ and angle bisector $AD$ ($D \in BC$). Denote by $\omega_1$ and $\omega_2$ the circumcircles of triangles $ABD$ and $ACD$, respectively. The line $AC$ intersects $\omega_1$ again at $F$ and the line $AB$ intersects $\omega_2$ again at $E$. The line $DE$ intersects $\omega_1$ again at $G$ and the line $DF$ intersects $\omega_2$ again at $H$. Prove that the circumcircles of triangles $ABC$, $AEF$ and $AGH$ have a second common intersection point.