Three months ago, I shared an original proof of Descartes's theorem here on my blog. Today it was finally published! It will be part of the december issue of The Mathematical Intelligencer, but you can already find it online. I made several improvements for clarity, so I consider that version to be definitive. If you don't have access though your library, you can read a preprint on ArXiV which is almost the same, just with a few more typos.