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prove that points A_3, B_3 and C_3 lie on a line

Source: 239MO 2004, grade 10-11, problem 5

December 11, 2004
geometrycircumcircleincenteranalytic geometryEulergeometry solved

Problem Statement

The incircle of triangle ABCABC touches its sides AB,BC,CAAB, BC, CA in points C1,A1,B1C_1, A_1, B_1 respectively. The point B2B_2 is symmetric to B1B_1 with respect to line A1C1A_1C_1, lines BB2BB_2 and ACAC meet in point B3B_3. points A3A_3 and C3C_3 may be defined analogously. Prove that points A3,B3A_3, B_3 and C3C_3 lie on a line, which passes through the circumcentre of a triangle ABCABC.
proposed by L. Emelyanov
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