The figure below shows a triangle ABC,
medians AD, BE,CF, and centroid G. O_{1}, O_{2}, O_{3}, O_{4}, O_{5}, O_{6}, are the circumcenters of
triangles AFG, AGE, BGF, BGD, CGD, and CGE, respectively. O_{1}O_{2} cuts O_{4}O_{3}
at H. HM is perpendicular to O_{2}O_{6} and HN is perpendicular to O_{4}O_{5}. Prove
that (1) HM = BE/2, HN = AD/2; (2) HO_{4}.HM = HO_{2}.HN
