I must be sleeping. Someone else will have to provide the easy way. This is what I did. It is BRUTE FORCE, to be sure.
For notational convenience, it's a few layers deep.
For simplicity, I thought we should do two things.
1) Set up coordinate axes so that fixed point on the circle is at the origin and the center is at (a,0) on the positive x-axis
2) Forget the bottom half of the circle. By symmetry, we have the same solution.
After that, define:
This defines all possible locations of the point chosen at random.
This is the distance from the origin for each point.
Further, calculating the probability directly, define:
Finally, we have
(Yes, this is tantalizingly close to a Reimann integral.)
I get a*1.273240 to about
My first few attempts continued to produce results that were independent of 'a'. Since that obviously is wrong, I had to look elsewhere. Obviously, I wandered around quite a bit.
For what it's worth. It is an interesting exercise.