This theorem relates the radii of four mutually tangent circles in the plane.

If three of the circles surround the fourth circle:

If three of the circles are [i]surrounded by[/i] 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 [url=https://www.geogebra.org/m/wy5tgvvd]six segment theorem[/url]. For more information, including some historical notes on the DCT, see [url=https://www.sangaku-journal.com/2024/SJM_2024_1-8_Unger.pdf]Notes on the Six-Segment Theorem[/url] by J. Marshall Unger.