Three points A, B, C lie on a circle centered at point O, so that the angle ABC is 60 degrees, the arc AB

Three points A, B, C lie on a circle centered at point O, so that the angle ABC is 60 degrees, the arc AB refers to the arc BC as 7: 5. Find the angles AOC, BAC, and ACB.

1. Angle ABC – inscribed, which means arc AC = 60 * 2 = 120
2. Angle AOC – central, so angle AOC = 120
3. Let’s compose and solve the equation:
7x + 5x + 120 = 360
12x = 240
x = 20
4. Arc AB = 20 * 7 = 140
Arc BC = 20 * 5 = 100
5. Angle BAC – inscribed, so angle BAC = 100: 2 = 50
Angle ACB – inscribed, so angle ACB = 140: 2 = 70
Answer: AOC angle = 120, BAC angle = 50, ACB angle = 70