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Locus with equilateral triangles

Source: Romania TST 2014 Day 1 Problem 3

December 25, 2014
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Problem Statement

Let A0A1A2A_0A_1A_2 be a scalene triangle. Find the locus of the centres of the equilateral triangles X0X1X2X_0X_1X_2 , such that AkA_k lies on the line Xk+1Xk+2X_{k+1}X_{k+2} for each k=0,1,2k=0,1,2 (with indices taken modulo 33).
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