SHARYGIN 2021 Problem 18

Let $ABC$ be a scalene triangle, $AM$ be the median through $A$ , and $\omega$ be the incircle. Let $\omega$ touch $BC$ at point $T$ and segment $AT$ meet $\omega$ for the second time at point $S$ . Let $\delta$ be the triangle formed by lines $AM$ and $BC$ and the tangent to $\omega$ at $S$ . Prove that the incircle of triangle $\delta$ is tangent to the circumcircle of triangle $ABC$ .