Question 1
Are the radii of the same circle OA, OB and OC are equal?
Yes! The radii of the same circle are always equal by definition. So, OA = OB = OC.
Proof Summary and Conclusion
Now, in an isosceles triangle OAC,
∠OAC + ∠OCA + ∠AOC = 180°
or, 2∠OCA = 180° - ∠AOC .......... (i) [∠OAC = ∠OCA]
Again, in an isosceles triangle OBC,
∠OCB + ∠OBC + ∠BOC = 180°
or, 2∠OCB = 180° - ∠BOC .......... (ii) [∠OCB = ∠OBC]
Adding equation (i) and (ii), we get
2(∠OCA + ∠OCB) = 360° - (∠AOC + ∠BOC)
or, 2∠ACB = 360° - Reflex ∠AOB
or, 2∠ACB = ∠AOB
or, 2∠ACB ≗ Arc AB
The relationship between the double the angle at the circumference and its opposite arc has the equal influence. It is denoted by 2∠ACB ≗ Arc AB