Triangle OPQ is isosceles since OP = OQ = rSo angle OPQ = angle OQP = a
Christiane Eaves,The Mount School York proved that the angle at the centre of a circle is twice the angle at the circumference subtended by the same arc, and it follows that the angles subtended at the circumference by the same arc are equal (i.e. angles in the same segment of a circle are equal).
Christiane proved that, for any points P, Q and R on a circle centre O,the angle at the centre ROP is twice the angle at the circumference RQP. It follows that, if PR is a fixed arc and Q is a variable point, then the angle PQR remains invariant.
Triangle OPQ is isosceles since OP = OQ = r
So angle OPQ = angle OQP = a
Using the angle sum of a triangle 2a + x = 180
But since angle POQ + angle TOP = 180 (angles on a straight
line)
then angle TOP = 2a
Similarly angle ROT = 2b.
It follows that angle ROP = 2a + 2b = 2(a + b) = 2 * angle RQP