If the average ray length is the average distance of the segments from a point inside the circle to points evenly distributed on the boundary:
Prove the center of the unit circle has the highest average ray length.
Convert the circle $x^2+y^2=1$ into polar coordinates so the distances are evenly distrbuted. Since distance $r$ is centered at the origin we must move point $(u,v)$ in the circle to the origin.
Solving for $r$ and simplifying give us
Since $r$ must be positive, we get the average radius is
Then solve the integral and find the maximum in terms of $(u,v)$
The problem is I'm not sure if the integral is solvable. Is there another way of approaching this problem?