SHARYGIN 2011 Problem 91

Altitudes $AA_1$ and $BB_1$ of triangle ABC meet in point $H$ . Line $CH$ meets the semicircle with diameter $AB$ , passing through $A_1, B_1$ , in point $D$ . Segments $AD$ and $BB_1$ meet in point $M$ , segments $BD$ and $AA_1$ meet in point $N$ . Prove that the circumcircles of triangles $B_1DM$ and $A_1DN$ touch.