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 link:
Clicking the link will download the file to view on your computer. To view the downloaded file you must have the Adobe Acrobat reader (or an equivalent program for viewing PDF files). Since PDF is a very widely used format for sharing documents, your computer likely already has the reader (you may even have a plug-in for your browser that lets it display PDF documents).