A circle with center O is inscribed in an equilateral triangle ABC. Find the degree measures of the angles AOC, AOB, BOC.

Since triangle ABC is equilateral, all of its internal angles are 60.

The center of a circle inscribed in a triangle is the point of intersection of the bisectors of the angles of the triangle.

Then the angle ОАС = 60/2 = 300, angle ОАС = 60/2 = 30.

Angle AOC = (180 – 30 – 30) = 120.

Similarly, the angle ОАВ = ОВА = ОВС = OCВ = 300, the angle AOB = BOS = 120.

Answer: The angles AOC, AOB, BOC are equal to 120.