Descartes' theorem ( externally tangent circle to three given kissing circles)


In geometry, Descartes’ theorem states that for every four kissing, or mutually tangent, circles, the radii of the circles satisfy a certain quadratic equation. By solving this equation, one can construct a fourth circle tangent to three given, mutually tangent circles. In mathematics, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line. Descartes’ theorem is most easily stated in terms of the circles’ curvatures. The curvature (or bend) of a circle is defined as k = ±1/r, where r is its radius. The larger a circle, the smaller is the magnitude of its curvature, and vice versa. If four circles are tangent to each other at six distinct points, and the circles have curvatures k1,k2,k3,k4 and trying to find the radius of the fourth circle that is externally tangent to three given kissing circles, Descartes’ theorem is giving the solution.

Related formulas


k4Curvature (or bend) of the externally tangent circle (m-1)
k1Curvature of the circle having radius r1 (m-1)
k2Curvature of the circle having radius r2 (m-1)
k3Curvature of the circle having radius r3 (m-1)