MathLabs

Problem 1

Six points are chosen on the sides of an equilateral triangle ABCABC: A1,A2A_1, A_2 on BCBC, B1,B2B_1, B_2 on CACA and C1,C2C_1, C_2 on ABAB, such that they are the vertices of a convex hexagon A1A2B1B2C1C2A_1A_2B_1B_2C_1C_2 with equal side lengths. Prove that the lines A1B2A_1B_2, B1C2B_1C_2 and C1A2C_1A_2 are concurrent.
Step 3 of 6: The remaining three sides also cancel
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In plain words

Removing a zero-sum triple from a zero total leaves another zero-sum triple.

A2B1→+B2C1→+C2A1→=0⃗\overrightarrow{A_2B_1}+\overrightarrow{B_2C_1}+\overrightarrow{C_2A_1}=\vec0
Detailed analysis

Subtracting the relation of the previous step from the six-term sum of the first step leaves A2B1→+B2C1→+C2A1→=0⃗\overrightarrow{A_2B_1}+\overrightarrow{B_2C_1}+\overrightarrow{C_2A_1}=\vec0. These three vectors also share the hexagon's common length, so — being equal-length vectors summing to zero — they too are related by successive 120∘120^\circ rotations.