Descartes' Theorem
If four circles are tangent to each other at six distinct points, and the circles have curvatures ki (for i = 1, ..., 4), Descartes' theorem says:
When trying to find the radius of a fourth circle tangent to three given kissing circles, the equation is best rewritten as:
The ± sign reflects the fact that there are in general two solutions. Ignoring the degenerate case of a straight line, one solution is positive and the other is either positive or negative; if negative, it represents a circle that circumscribes the first three (as shown in the diagram above).
Other criteria may favor one solution over the other in any given problem.
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“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)