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 O_{1}
and diameter HE cuts circle O
at F. If O_{2} is the circumcenter of triangle FDM, prove that (1)
Points F, O_{1}, and O_{2} are collinear; (2) Circles O_{2} and O_{1} are
tangent at F.