Baltic Way 2023 Shortlist

In an acute triangle $△ABC$ with $|AB|\ne |AC|$, the angle bisector of $\angle BAC$ intersects side $BC$ and $\odot (ABC)$ at the points $D$ and $M_A$, respectively. Let points $X$ and $Y$ be the feet of perpendiculars from $M_A$ to sides $AB$ and $AC$, respectively. The tangent of $\odot (BX{M}_A)$ at the point $X$ and the tangent of $\odot (CY{M}_A)$ at the point $Y$ intersect at the point $T$. Suppose that lines $AT$ and $BC$ intersect at the point $S$. Show that $\odot (TS{M}_A)$ passes through the midpoint of segment $AD$.