Problem 1

Six points are chosen on the sides of an equilateral triangle

ABC

A_1, A_2

BC

B_1, B_2

CA

and

C_1, C_2

AB

, such that they are the vertices of a convex hexagon

A_1A_2B_1B_2C_1C_2

with equal side lengths. Prove that the lines

A_1B_2

B_1C_2

and

C_1A_2

are concurrent.

Step 2 of 6: Three equal-length sides cancel

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In plain words

Equal-length vectors pointing along the three directions of an equilateral triangle, each rotated $120^\circ$ from the last, always add to zero.

\overrightarrow{A_1A_2}+\overrightarrow{B_1B_2}+\overrightarrow{C_1C_2}=\vec0

Detailed analysis

The segments $A_1A_2$ , $B_1B_2$ , $C_1C_2$ all have the same length (the hexagon's common side length) and lie along $BC$ , $CA$ , $AB$ respectively, directions that differ by successive $120^\circ$ rotations around the equilateral triangle $ABC$ . Three equal-length vectors related by $120^\circ$ rotations sum to $\vec0$ .