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.