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a configuration with gergonne point and its cevians

Source: Indonesia TST 2016 Round 3

March 24, 2024
geometryincentergergonne

Problem Statement

In a non-isosceles triangle ABCABC, let II be its incenter. The incircle of ABCABC touches BCBC, CACA, and ABAB at DD, EE, and FF, respectively. A line passing through DD and perpendicular to ADAD intersects IBIB and ICIC at AbA_b and AcA_c, respectively. Define the points BcB_c, BaB_a, CaC_a, and CbC_b similarly. Let GG be the intersection of the cevians ADAD, BEBE, and CFCF. The points O1O_1 and O2O_2 are the circumcenter of the triangles AbBcCaA_bB_cC_a and AcBaCbA_cB_aC_b, respectively. Prove that IGIG is the perpendicular bisector of O1O2O_1O_2.
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