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Prove that 3 tangents are concurrent

Source: 2019 Brazil IMO TST 3.2

June 13, 2023
geometryconcurrencyEquilateral Triangletangentexternal tangent

Problem Statement

Let ABCABC be a triangle, and A1A_1, B1B_1, C1C_1 points on the sides BCBC, CACA, ABAB, respectively, such that the triangle A1B1C1A_1B_1C_1 is equilateral. Let I1I_1 and ω1\omega_1 be the incenter and the incircle of AB1C1AB_1C_1. Define I2I_2, ω2\omega_2 and I3I_3, ω3\omega_3 similarly, with respect to the triangles BA1C1BA_1C_1 and CA1B1CA_1B_1, respectively. Let l1≠BCl_1 \neq BC be the external tangent line to ω2\omega_2 and ω3\omega_3. Define l2l_2 and l3l_3 similarly, with respect to the pairs ω1\omega_1, ω3\omega_3 and ω1\omega_1, ω2\omega_2.
Knowing that A1I2=A1I3A_1I_2 = A_1I_3, show that the lines l1l_1, l2l_2, l3l_3 are concurrent.
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