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Descartes' circle theorem 2026-10-20

Four mutually tangent circles with signed curvatures

k_{i}

satisfy

(k_{1} + k_{2} + k_{3} + k_{4})^{2} = 2 (k_{1}^{2} + k_{2}^{2} + k_{3}^{2} + k_{4}^{2})

. Curvature is the reciprocal of the radius, with an enclosing circle assigned negative curvature. Given three curvatures, solving the quadratic equation gives

k_{4} = k_{1} + k_{2} + k_{3} \pm 2 k_{1} k_{2} + k_{2} k_{3} + k_{3} k_{1}

, providing the two completions of the configuration.

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