Inversion III: Abstract

We introduce the radical axis of two circles and inverstigate its properties. We prove that there is an inversion which sends two non-intersecting circles into two concentric cicrles, thus solving the famous "ring of circles between two circles" problem. We also apply the fact that the three radical axes of three circles are either parallel or intersect in a point. We use this and other properties of radical axes to solve advanced Olympiad-style problems.