Assume two adjacent circles as in this figure:
Given distances $r,l,d$ and the angle $\theta$, how to find the expected distance between point $A$ in the left circle and a uniform random point in right circle. (parametric/analytical approach)
Extended version:
What would the solution be if $d$ was known and $\theta$ was unknown. That is, $A$ can be any point on the edge of a circle with radius $d$ where this circle shares the same center with the left circle in the figure.
Hint: Calculate the radius of a smallest circle with center at $A$ that still touches the smaller circle, $r_{-}$. Also, calculate the radius of a largest circle with center at $A$ that still touches the smaller circle, $r_{+}$. Then, for any radius $r\in[r_{-},r_{+}]$, calculate the arc-length of intersection of the smaller circle with the circle with radius $r$ and center at at $A$, $a(r)$. This is pdf over which you need to integrate $r$ to get your answer.