Coming full circle

The purpose of this math circle was to provide a gentle yet reasonably complete introduction to the concept of inversion. We developed much of the theory through a series of problems, outlined below. You will need to be familiar with similar triangles and the interplay between angles and circles, but otherwise not much geometric background is required. (To understand the basics more quickly, skip the problems marked ‘Optional’ or ‘Challenge.’) If you have questions or solutions that you would like to share, please send them to me at samv@math.stanford.edu at any point.