Online Geometry Problem 843: Triangle, Medians, Centroid, Four Circumcenters, Perpendicular, Parallel, Congruence, Similarity. Level: High School, SAT Prep, College, Mathematics Education

The figure below shows a triangle ABC, medians AD, BE, CF, and centroid G. O₁, O₂, O₃, O₄ are the circumcenters of triangles AFG, AGE, BGF, and BGD, respectively. O₁O₂ cuts O₄O₃ at H. CP is parallel to AD (P on BE extended). Prove that HO₃.CP = HO₁.GP.