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 4 of 6: A circle through the two cut points

Previous step Next step

In plain words

Two points that see the same segment at the same angle, on the same side, lie on a common circle through its endpoints.

\angle A_1DB_1=\angle A_1CB_1=60^\circ,\quad D=C_2A_1\cap A_2B_1

Detailed analysis

Let $D=C_2A_1\cap A_2B_1$ . The $120^\circ$ relation of the previous step makes $\angle A_1DB_1=60^\circ$ , and $\angle A_1CB_1=60^\circ$ simply because it is the angle of the equilateral triangle at $C$ . Equal angles subtending $A_1B_1$ from the same side put $A_1,B_1,C,D$ on one circle.