Question 151
Let $\omega$ be the circle passing through point $C$ and tangent to side $AB$ of triangle $ABC$ at point $B$. The interior bisector of angle $\angle A$ intersects circle $\omega$ at points $E$ and $F$, and side $BC$ at point $D$ such that $F$ lies inside triangle $ABC$. Point $S$ on segment $EC$ is chosen such that $2\angle EBS = \angle BAC$. Let $T$ be a point on segment $BS$ such that line $DT$ is parallel to $CS$. Line $ET$ intersects circle $\omega$ again at point $U$. Prove that the circumcircle of triangle $ABU$ is tangent to line $AD$.
(Khulan Tumenbayar)
(Khulan Tumenbayar)