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Line through O given concurrency

Source: 2024 Israel Olympic Revenge P4

June 20, 2024
geometryolympic revenge

Problem Statement

Let ABCABC be an acute triangle. Let DD be a point inside side BCBC. Let EE be the foot from DD to ACAC, and let FF be a point on ABAB so that FE⊥ABFE\perp AB. It is given that the lines AD,BE,CFAD, BE, CF concur. MA,MB,MCM_A, M_B, M_C are the midpoints of sides BC,AC,ABBC, AC, AB respectively, and OO is the circumcenter of ABCABC. Moreover, we define P=EF∩MAMB,S=DE∩MAMCP=EF\cap M_AM_B, S=DE\cap M_AM_C. Prove that O,P,SO, P, S are collinear.
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