In an acute-angled triangle ABC, points A, C, the center of the inscribed circle O

In an acute-angled triangle ABC, points A, C, the center of the inscribed circle O and the center of the inscribed circle I lie on the same circle. Prove that the angle ABC is 60º.

Since point O is the center of the circumscribed circle, angle AOC is central, and angle B is inscribed. In triangle ABC we have: angle AOC = 2 angles ABC, and angle AIC = 180 (degrees) – angle ABC. Therefore: 2 angles ABC = 180 (degrees) – angle ABC, this means that angle ABC = 60 (degrees)