SHARYGIN 2023 Problem 15

Let $ABCD$ be a convex quadrilateral. Points $X$ and $Y$ lie on the extensions beyond $D$ of the sides $CD$ and $AD$ respectively in such a way that $DX = AB$ and $DY = BC$ . Similarly points $Z$ and $T$ lie on the extensions beyond $B$ of the sides $CB$ and $AB$ respectively in such a way that $BZ = AD$ and $BT = DC$ . Let $M_1$ be the midpoint of $XY$ , and $M_2$ be the midpoint of $ZT$ . Prove that the lines $DM_1, BM_2$ and $AC$ concur.