Circle packing theorem is a famous result stating that for every connected simple planar graph $G$ there is a circle packing in the plane whose intersection graph is $G$ https://en.wikipedia.org/wiki/Circle_packing_theorem.

I know that this result has many proofs and I want to read one of them, but don't understand how to start (for quite a while). The article in wiki gives a reference to Thurston notes, but the proof comes only in the last section and I am not sure if this is the simplest approach. I like these notes very much, but was never able to read them till the end. So I wonder if there are some simple proofs of this result nowadays. Can you advise something?

  • $\begingroup$ may be the fact that there exists a triangulation of every maximal planar graph may come handy in the proof $\endgroup$
    – vidyarthi
    Aug 25, 2020 at 10:28
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    $\begingroup$ Does this answer to your question?mathoverflow.net/q/187845/90655 $\endgroup$
    – C.F.G
    Sep 3, 2020 at 8:23
  • $\begingroup$ Thanks a lot C.F.G! I have not spotted this question. It looks like mine is a duplicate. I'll study the answers $\endgroup$
    – aglearner
    Sep 3, 2020 at 13:39
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    $\begingroup$ Although your question is close to a duplicate to "Koebe–Andreev–Thurston theorem - where can I find a proof?," additional expositions have appeared in the ~6 yrs since that post. $\endgroup$ Sep 3, 2020 at 17:26

2 Answers 2


I can recommend Sariel Har-Peled's exposition in supplemental Chapter 15 of his book Geometric Approximation Algorithms. Ch15 PDF download. He emphasizes angles via a "whac-an-angle" game. He acknowledges that

Our presentation follows Pach and Agarwal [pa-cg-95].


Books are written on the subject, so, finding a proof (which are many by now) shouldn't be a problem. I also enjoyed greatly Rohde's tribute to Schramm that explains in very nice way some ideas that Schramm introduced into the area; following references from there one should be able to find more detailed accounts.


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