Awesome, thanks so much for putting that talented cranium to work for me.
That's fascinating - I didn't know of the inscribed angle law. That totally was the keystone for this problem.
In the interests of me not having to ask the same questions at a later date, would you mind if I paraphrase my understanding of this, and hopefully if you could correct me where my assertions are incorrect? It would help me solidify this in my head.
As I understand it, you're building a circle that inscribes an angle of theta_AB and reaches out to two points that are |AB| apart.
You can then find the 'height' from the mid point of the |AB| chord to the inner point of the circle using simple tan = opposite/adjacent, solving for the 'adjacent', and the radius using a similar sin = opposite/hypotenuse, solving for hypotenuse.
Man, writing that down even makes me think of it clearer. I hope I'm not wrong in my paraphrasing, as that sounds more and more correct.
After that, you're right - treating the original orbit as simply a circle that intersects with this larger circle should make solving this solution much simpler. Off the top of my head I don't remember calculating circle intersections, but I remember doing it in junior high, so I can't imagine it'll be too hard to scrape the rust off the noggin.
Thanks again Reed! Please let me know if I've thought of any of this incorrectly.
P.S. If you don't mind, how did you end up drawing such good math diagrams? Mine was just a botched MS Paint job, yours looks much better.