MathDB
3 parallel lines + collinear related to intersections of circumcircles

Source: Sharygin 2011 Final 9.6

December 20, 2018
geometrycircumcirclealtitudecollinearcollinearityParallel Lines

Problem Statement

In triangle ABCABC AA0AA_0 and BB0BB_0 are medians, AA1AA_1 and BB1BB_1 are altitudes. The circumcircles of triangles CA0B0CA_0B_0 and CA1B1CA_1B_1 meet again in point McM_c. Points Ma,MbM_a, M_b are defined similarly. Prove that points Ma,Mb,McM_a, M_b, M_c are collinear and lines AMa,BMb,CMcAM_a, BM_b, CM_c are parallel.