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An ugly geo in Cono Sur

Source: Cono Sur 2024 P2

September 27, 2024
geometryvectorbarycentric coordinatescomplex numbers

Problem Statement

Let ABCABC be a triangle. Let A1A_1 and A2A_2 be points on side BC,B1BC, B_1 and B2B_2 be points on side CACA and C1C_1 and C2C_2 be points on side ABAB such that A1A2B1B2C1C2A_1A_2B_1B_2C_1C_2 is a convex hexagon and that B,A1,A2B,A_1,A_2 and CC are located in that order on side BCBC. We say that triangles AB2C1,BA1C2AB_2C_1, BA_1C_2 and CA2B1CA_2B_1 are glueable if there exists a triangle PQRPQR and there exist X,YX,Y and ZZ on sides QR,RPQR, RP and PQPQ respectively, such that triangle AB2C1AB_2C_1 is congruent in that order to triangle PYZPYZ, triangle BA1C2BA_1C_2 is congruent in that order to triangle QXZQXZ and triangle CA2B1CA_2B_1 is congruent in that order to triangle RXYRXY. Prove that triangles AB2C1,BA1C2AB_2C_1, BA_1C_2 and CA2B1CA_2B_1 are glueable if and only if the centroids of triangles A1B1C1A_1B_1C_1 and A2B2C2A_2B_2C_2 coincide.
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