Descartes' Theorem - Complex Descartes Theorem

Complex Descartes Theorem

To determine a circle completely, not only its radius (or curvature), but also its center must be known. The relevant equation is expressed most clearly if the coordinates (x, y) are interpreted as a complex number z = x + iy. The equation then looks similar to Descartes' theorem and is therefore called the complex Descartes theorem.

Given four circles with curvatures k_i and centers z_i (for i = 1...4), the following equality holds in addition to equation (1):

Once k₄ has been found using equation (2), one may proceed to calculate z₄ by rewriting equation (4) to a form similar to equation (2):

Again, in general, there are two solutions for z₄, corresponding to the two solutions for k₄.