Descartes' circle theorem

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 $k_4=k_1+k_2+k_3\pm 2\sqrt{k_1{k}_2+k_2{k}_3+k_3{k}_1}$, providing the two completions of the configuration.