Geometry Problem 1137: Triangle, Circumcircle, Orthocenter, Midpoint, Arc, Collinear Points, Tangent Circles. Level: High School, SAT Prep, Honors Geometry, College, Math Education

In a triangle ABC (figure below) with circumcircle O, altitude BD, and orthocenter H, M is the midpoint of AC. MH extended cuts the arc BC at E. Circle of center O₁ and diameter HE cuts circle O at F. If O₂ is the circumcenter of triangle FDM, prove that (1) Points F, O₁, and O₂ are collinear; (2) Circles O₂ and O₁ are tangent at F.