IGO 2022 A2

We are given an acute triangle $ABC$ with $AB\neq AC$ . Let $D$ be a point of $BC$ such that $DA$ is tangent to the circumcircle of $ABC$ . Let $E$ and $F$ be the circumcenters of triangles $ABD$ and $ACD$ , respectively, and let $M$ be the midpoints $EF$ . Prove that the line tangent to the circumcircle of $AMD$ through $D$ is also tangent to the circumcircle of $ABC$ .

Proposed by Patrik Bak, Slovakia