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Coincideness of two incenters

Source: 2023 Turkey Egmo Tst P1

March 23, 2023
geometryincentercircumcircle

Problem Statement

Let O1O2O3O_1O_2O_3 be an acute angled triangle.Let ω1\omega_1, ω2\omega_2, ω3\omega_3 be the circles with centres O1O_1, O2O_2, O3O_3 respectively such that any of two are tangent to each other. Circumcircle of O1O2O3O_1O_2O_3 intersects ω1\omega_1 at A1A_1 and B1B_1, ω2\omega_2 at A2A_2 and B2B_2, ω3\omega_3 at A3A_3 and B3B_3 respectively. Prove that the incenter of triangle which can be constructed by lines A1B1A_1B_1, A2B2A_2B_2, A3B3A_3B_3 and the incenter of O1O2O3O_1O_2O_3 are coincide.
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