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Concurrence of three radical axes

Source: Kürschak 2010, problem 2

July 8, 2014
trigonometrygeometry unsolvedgeometry

Problem Statement

Consider a triangle ABCABC, with the points A1A_1, A2A_2 on side BCBC, B1,B2∈AC‾B_1,B_2\in\overline{AC}, C1,C2∈AB‾C_1,C_2\in\overline{AB} such that AC1<AC2AC_1<AC_2, BA1<BA2BA_1<BA_2, CB1<CB2CB_1<CB_2. Let the circles AB1C1AB_1C_1 and AB2C2AB_2C_2 meet at AA and A∗A^*. Similarly, let the circles BC1A1BC_1A_1 and BC2A2BC_2A_2 intersect at B∗≠BB^*\neq B, let CA1B1CA_1B_1 and CA2B2CA_2B_2 intersect at C∗≠CC^*\neq C. Prove that the lines AA∗AA^*, BB∗BB^*, CC∗CC^* are concurrent.
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