First off, sorry for the clumsy title. I have no idea what this problem should be called. Long story short, I'm working on some software that simulates satellites that visually monitor space junk in orbit, and I need to calculate the maximum distance from one circular orbit to another that can be completely covered by the satellites' cameras if I have n satellites equally spaced with and an effective range d that always results in an overlap.
To simplify, if I have two small, overlapping circles drawn with their center points on a larger circle as shown in this diagram, how can I find the distance from the center of the larger circle to the points of intersection of the satellites' effective areas as shown by the red circles in this diagram. I'm trying to find the two circles shown in red that pass through points E and F. Also, what are some good search terms for this kind of problem? I will eventually need to do more complex (realistic) versions of this problem.
Let $G$ be the midpoint of both $\overline{CD}$ and $\overline{EF}$, and define $r = AD$ and $\theta = m\angle DAG = \pi/n$. Then $AG = r \cos \theta$ and $DG = r\sin \theta$. The side lengths of the right triangle $\triangle DGE$ satisfy $DG^2 + EG^2 = DE^2$, so $EG^2 = d^2 - r^2 \sin^2 \theta$. Therefore \begin{equation} AE = r \cos \theta - \sqrt{d^2 - r^2 \sin^2 \theta}, \quad \text{and} \quad AF = r \cos \theta + \sqrt{d^2 - r^2 \sin^2 \theta}. \end{equation}