KöMaL 2021 B5148

A triangle $ABC$ is right angled at $C$. The inscribed circle touches the leg $BC$ at point $D$, and the leg $AC$ at point $E$. The escribed circle of side $BC$ touches line segment $BC$ at point $G$; and the escribed circle of side $AC$ touches line segment $AC$ at point $H$. The intersection of line segments $DH$ and $EG$ is $M$. Show that the other intersection of the circumscribed circles of triangles $DGM$ and $EHM$ lies on the inscribed circle.