American Mathematical Monthly
We present a short proof of Descartes Circle Theorem on the “curvature-centers” of four mutually tangent circles. Key to the proof is associating an octahedral configuration of spheres to four mutually tangent circles. We also prove an analogue for spheres.
Northshield, S. (2014). Complex Descartes Circle Theorem. American Mathematical Monthly, 121 (10), 927-931.