What about using a fourth point 'd' on the circle and below the middle point 'o' between a and b, such that angle dca = dcb, and such that the circle with center at 'm' would have a radius r = ma = mb = mc = md.
Together with known angle acb and knowing that angle dca = dcb. Isn't that somehow enough info to get r in relationship to the distance of the chord ab?
But note that the points 'c', 'o' and 'd' are not on a straight line. 'd' is instead defined from angles and radius and on a straight line with 'o' and circle center point 'm'
m (m is circle center)
a o b