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 S_{1} = area of triangle AEG, S_{2}= area of triangle CFM, and
S_{3} = area of triangle DGM, prove that S_{1} + S_{2} = S_{3}.
