Descartes' Circle Theorem
This theorem relates the radii of four mutually tangent circles in the plane.
Version 1
If three of the circles surround the fourth circle:
Version 2
If three of the circles are surrounded by the fourth circle (note the sign change):
A version of the Descartes' Circle Theorem (DCT) was written down by a Japanese mathematician in 1751. It can be proven readily by applying the six segment theorem. For more information, including some historical notes on the DCT, see Notes on the Six-Segment Theorem by J. Marshall Unger.