On a circle with center O, points A, B, C are taken sequentially so that the angle AOB is equal to the angle BOC

On a circle with center O, points A, B, C are taken sequentially so that the angle AOB is equal to the angle BOC is equal to the angle COA. Is it true that triangle ABC is equilateral?

Yes, that’s right, triangle ABC is equilateral.

Solution: Consider which arcs are supported by the angles AOB, BOC and COA (all angles are central angles): angle AOB rests on arc AB, angle BOC rests on arc BC, angle SOA rests on arc AC. Consequently, the arc AB is equal to the arc BC and is equal to the arc AC.

If you draw the segments AB, BC and AC (they will be the chords of the given circle), then the chords resting on equal arcs are equal to each other.

Therefore, AB = BC = AC, that is, the triangle ABC = equilateral.