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Circumcenter of incenter triangle

Source: Saudi Arabia IMO TST Day IV Problem 4

July 22, 2014
geometrycircumcircleincentergeometric transformationreflectionperpendicular bisectorangle bisector

Problem Statement

Points A1, B1, C1A_1,~ B_1,~ C_1 lie on the sides BC, ACBC,~ AC and ABAB of a triangle ABCABC, respectively, such that AB1−AC1=CA1−CB1=BC1−BA1AB_1 -AC_1 = CA_1 -CB_1 = BC_1 -BA_1. Let IA, IB, ICI_A,~ I_B,~ I_C be the incenters of triangles AB1C1, A1BC1AB_1C_1,~ A_1BC_1 and A1B1CA_1B_1C respectively. Prove that the circumcenter of triangle IAIBICI_AI_BI_C, is the incenter of triangle ABCABC.
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