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Equilateral pair

Source: RMO 2019 P2

October 20, 2019
geometryRMOP2

Problem Statement

Let ABCABC be a triangle with circumcircle Ω\Omega and let GG be the centroid of triangle ABCABC. Extend AG,BGAG, BG and CGCG to meet the circle Ω\Omega again in A1,B1A_1, B_1 and C1C_1. Suppose ∠BAC=∠A1B1C1,∠ABC=∠A1C1B1\angle BAC = \angle A_1 B_1 C_1, \angle ABC = \angle A_1 C_1 B_1 and ∠ACB=B1A1C1 \angle ACB = B_1 A_1 C_1. Prove that ABCABC and A1B1C1A_1 B_1 C_1 are equilateral triangles.
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