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Lines connecting centroids

Source: 2023 Taiwan TST Round 1 Independent Study 1-G

March 15, 2023
Taiwangeometry

Problem Statement

Let ABCABC be a triangle. Let ABC1,BCA1,CAB1ABC_1, BCA_1, CAB_1 be three equilateral triangles that do not overlap with ABCABC. Let PP be the intersection of the circumcircles of triangle ABC1ABC_1 and CAB1CAB_1. Let QQ be the point on the circumcircle of triangle CAB1CAB_1 so that PQPQ is parallel to BA1BA_1. Let RR be the point on the circumcircle of triangle ABC1ABC_1 so that PRPR is parallel to CA1CA_1.
Show that the line connecting the centroid of triangle ABCABC and the centroid of triangle PQRPQR is parallel to BCBC.
Proposed by usjl
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