I'm interested in making a program that draws the Apollonian gasket fractal. To do this, I need a method to find the center of a circle internally or externally tangent to three Soddy circles. Finding the radius is easy enough to do using Descartes' theorem, but there aren't any methods to find the circle's center. I am currently doing it using trilateration, since the distances to any given Soddy circle is the sum of its radius and the central circle's radius. Are there any simpler approaches to do this?
Edit: I'm actually using this algebraic approach to find the fourth Soddy circle, which works fine for three externally tangent circles. Jean Marie's comment links to a question about a complex version of Descartes' theorem, which is all well and good, but how do I solve the equation and choose a root?