Descartes' circle theorem
= Descartes' circle theorem
{c}
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 $k_4=k_1+k_2+k_3\pm2\sqrt{k_1k_2+k_2k_3+k_3k_1}$, providing the two completions of the configuration.