Descartes Complex Circle Theorem
This is a sketch of the setup for proving the Complex Version of Descartes' Circle Theorem.
See http://digitalcommons.plattsburgh.edu/cgi/viewcontent.cgi?article=1000&context=mathematics_facpubs
(Published as Northshield, S. (2014). Complex Descartes Circle Theorem. American Mathematical Monthly, 121 (10), 927-931.)
See http://tube.geogebra.org/student/m1294113 for a version that includes an algebra window that you can use to show or hide points and spheres.