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Bob drawing three triangles, wants to control their perimeter

Source: Baltic Way 2020, Problem 15

November 14, 2020
geometrygeometry proposeddistancesperimeter

Problem Statement

On a plane, Bob chooses 3 points A0A_0, B0B_0, C0C_0 (not necessarily distinct) such that A0B0+B0C0+C0A0=1A_0B_0+B_0C_0+C_0A_0=1. Then he chooses points A1A_1, B1B_1, C1C_1 (not necessarily distinct) in such a way that A1B1=A0B0A_1B_1=A_0B_0 and B1C1=B0C0B_1C_1=B_0C_0. Next he chooses points A2A_2, B2B_2, C2C_2 as a permutation of points A1A_1, B1B_1, C1C_1. Finally, Bob chooses points A3A_3, B3B_3, C3C_3 (not necessarily distinct) in such a way that A3B3=A2B2A_3B_3=A_2B_2 and B3C3=B2C2B_3C_3=B_2C_2. What are the smallest and the greatest possible values of A3B3+B3C3+C3A3A_3B_3+B_3C_3+C_3A_3 Bob can obtain?
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