tangent circles

[ghenghea]

Let $ABC$ be a triangle with incircle $\omega$, and let $T$ be the tangency point of the $A$-mixtilinear incircle with the circumcircle. Let $T'$ denote the reflection of $T$ over $\overline{BC}$. Prove that $\omega$ is internally tangent to the nine-point circle of triangle $BT'C$.