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 5 of 6: One vertex, two equal chords

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In plain words

Chasing angles around the circle through $A_1,B_1,C,D$ turns the equal inscribed angles into two congruent triangles at $A_2$ .

A_2B_2=A_2C_2

Detailed analysis

Equal inscribed angles in that circle give $\angle CA_1D=\angle CB_1D$ , hence $\angle C_2A_1A_2=\angle A_2B_1B_2$ . Together with the hexagon's equal side lengths this makes triangles $C_2A_1A_2$ and $A_2B_1B_2$ congruent, so $A_2C_2=A_2B_2$ : vertex $A_2$ is equidistant from $B_2$ and $C_2$ .