Proofs of The Theorem
There are many known proofs of the circle packing theorem. Paul Koebe's original proof is based on his conformal uniformization theorem saying that a finitely connected planar domain is conformally equivalent to a circle domain. There are several different topological proofs that are known. Thurston's proof is based on Brouwer's fixed point theorem. There is also a proof using a discrete variant of Perron's method of constructing solutions to the Dirichlet problem. Yves Colin de Verdière proved the existence of the circle packing as a minimizer of a convex function on a certain configuration space.
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