Talk:Monge's theorem
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[edit]Hi Friends. Take a look in this file. Chears Rossi pena (talk) 16:48, 4 November 2012 (UTC)
See Also
[edit]I think the problem of Apollonius may be a stretch for this section. What makes it and Monge's theorem 'special' as results about three circles in the plane (e.g radical centre is another result in this spirit)? I could find this that links the two, but 'Monge' cannot be found on the Problem of Apollonius wiki page. --Hu5k3rDu (talk) 17:30, 7 March 2018 (UTC)
Error
[edit]The "simple proof" given in the article is in fact incorrect, as pointed out to me by Prof. Jerome Lewis of U. of South Carolina Upstate. The problem is that there may be no plane tangent to the three spheres (imagine, for example, that one of them is much smaller and lies between the other two). This flaw is correctable in several ways, among them by building cones instead of spheres on top of the original circles; you can find that proof, attributed to Nathan Bowler of Trinity College, Cambridge, on p. 95 of my book "Mathematical Mindbenders." ---Peter Winkler 3/27/18 — Preceding unsigned comment added by 129.170.28.210 (talk) 16:04, 27 March 2018 (UTC)
Distinct radii
[edit]Should the introduction mention that the three circles must have distinct radii for this theorem to apply? Otherwise the pair of tangent lines never intersect. --Maxime Petazzoni 04:37, 17 July 2018 (UTC)
Further cases?
[edit]"Each pair of spheres defines a cone that is externally tangent to both spheres". Actually, by going to 3 dimensions each pair of spheres will define two pairs of such cones (because the usual internal tangents of a pair of circles now accounts for one of these in addition to its external pair.) In this sense there are three further points of intersection of two planes - only different planes in each new case and - going by symmetry of the sythetic proof already given - there will be three additional sets of collinearities and three additional planes that are mutally tangent to the spheres. 11:42, 23 September 2018 (UTC) — Preceding unsigned comment added by 2.26.222.1 (talk)