The Descartes' circle formula

Descartes' circle formula relates the curvatures (the inverse of the radius) of four mutually tangent circles. If these four circles all touch each other without overlaps, and their curvatures are \(k_1, k_2, k_3,\) and \(k_4\), then the formula states that: \[ (k_1 + k_2 + k_3 + k_4)^2 = 2(k_1^2 + k_2^2 + k_3^2 + k_4^2). \] This relationship helps determine the size and placement of circles that fit together perfectly, such as in complex packing or tangent-circle problems, providing a way to calculate one circle’s curvature based on the others.