Question 34
In $\triangle ABC$, let $D$ be a point on side $BC$. Suppose the incircle $\omega_1$ of $\triangle ABD$ touches sides $AB$ and $AD$ at $E, F$ respectively, and the incircle $\omega_2$ of $\triangle ACD$ touches sides $AD$ and $AC$ at $F, G$ respectively. Suppose the segment $EG$ intersects $\omega_1$ and $\omega_2$ again at $P$ and $Q$ respectively. Show that line $AD$, tangent of $\omega_1$ at $P$ and tangent of $\omega_2$ at $Q$ are concurrent.