The cyclic octagon ABCDEFGH has sides a, a, a, a, b, b, b, b respectively. Find the radius
of the circle that circumscribes ABCDEFGH in terms of a and b.

Well sides a and b between them subtend 90 degrees. If you drop a perpendicular from the center to side a, then the radius (call it r) is the hypotenuse of a right triangle whose far side is a/2. The angle that a subtends is therefore 2*arcsin(a/2r). Therefore, 2*arcsin(a/2r) + 2*arcsin(b/2r) = 90 degrees. That's as far as I got.