PL-OM 2017 Problem 5

Point $M$ is the midpoint of $BC$ of a triangle $ABC$ , in which $AB=AC$ . Point $D$ is the orthogonal projection of $M$ on $AB$ . Circle $\omega$ is inscribed in triangle $ACD$ and tangent to segments $AD$ and $AC$ at $K$ and $L$ respectively. Lines tangent to $\omega$ which pass through $M$ cross line $KL$ at $X$ and $Y$ , where points $X$ , $K$ , $L$ and $Y$ lie on $KL$ in this specific order. Prove that points $M$ , $D$ , $X$ and $Y$ are concyclic.