Geometry Problem 729: Triangle, Altitude, Orthocenter, Circumcircle, Perpendicular, Midpoint. Level: High School, Honors Geometry, College, Mathematics Education

The diagram shows a triangle ABC with the orthocenter H and the circumcircle O. D is a point on the arc BC, DE is perpendicular to AB, and DF is perpendicular to BC. DH meets EF at M. Prove that M is the midpoint of DH.