Let ABC be a triangle. Let A1 and A2 be points on side BC,B1 and B2 be points on side CA and C1 and C2 be points on side AB such that A1A2B1B2C1C2 is a convex hexagon and that B,A1,A2 and C are located in that order on side BC.
We say that triangles AB2C1,BA1C2 and CA2B1 are glueable if there exists a triangle PQR and there exist X,Y and Z on sides QR,RP and PQ respectively, such that triangle AB2C1 is congruent in that order to triangle PYZ, triangle BA1C2 is congruent in that order to triangle QXZ and triangle CA2B1 is congruent in that order to triangle RXY. Prove that triangles AB2C1,BA1C2 and CA2B1 are glueable if and only if the centroids of triangles A1B1C1 and A2B2C2 coincide. geometryvectorbarycentric coordinatescomplex numbers