This blog concerns how to determine the intersection between two circles in the plane algebraically. It is a problem that can crop up in a variety of situations, from gaming to tools for computer aided design to astronomy. This problem is interesting because is it a conceptually simple problem whose algebraic formulation is nonetheless apparently complex: a system of non-linear equations that are quadratic in both variables. However, by doing some geometric analysis of the problem, and applying tools from vector geometry, we are lead to a specific mathematical transformation of the problem that radically simplifies it. The key idea turns out to be a specific change of basis. My exposition of this uses a few diagrams (which are not supported by the blog editor) and a lot of mathematical expressions (which are clumsy to create in the blog editor), so I put it in an Adobe Portable Document Format (PDF) document that you can access using the following... read more