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Circle passes through the circumcenter

Source: Sharygin contest. The final raund. 2008. Grade 9. First day. Problem 4

August 31, 2008
geometrycircumcircleangle bisectorgeometry unsolved

Problem Statement

(F.Nilov, A.Zaslavsky) Let CC0 CC_0 be a median of triangle ABC ABC; the perpendicular bisectors to AC AC and BC BC intersect CC0 CC_0 in points Ac A_c, Bc B_c; C1 C_1 is the common point of AAc AA_c and BBc BB_c. Points A1 A_1, B1 B_1 are defined similarly. Prove that circle A1B1C1 A_1B_1C_1 passes through the circumcenter of triangle ABC ABC.
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