The figure below shows a triangle ABC,
medians AD, BE, CF, and centroid G. O_{1}, O_{2}, O_{3}, O_{4} are the circumcenters of
triangles AFG, AGE, BGF, and BGD, respectively. O_{1}O_{2} cuts O_{4}O_{3}
at H. CP is parallel to AD (P on BE extended). Prove
that HO_{3}.CP = HO_{1}.GP.
