In a triangle ABC,
BH is the altitude and BD is a diameter of the circumcircle O (see the figure below).
HE and HF are perpendicular
to AB and BC at E and F, respectively. DE and DF meet AC at G and M,
respectively. If S1 = area of triangle AEG, S2= area of triangle CFM, and
S3 = area of triangle DGM, prove that S1 + S2 = S3.