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- Problem_of_Apollonius abstract "In Euclidean plane geometry, Apollonius's problem is to construct circles that are tangent to three given circles in a plane (Figure 1). Apollonius of Perga (ca. 262 BC – ca. 190 BC) posed and solved this famous problem in his work Ἐπαφαί (Epaphaí, "Tangencies"); this work has been lost, but a 4th-century report of his results by Pappus of Alexandria has survived. Three given circles generically have eight different circles that are tangent to them (Figure 2) and each solution circle encloses or excludes the three given circles in a different way: in each solution, a different subset of the three circles is enclosed (its complement is excluded) and there are 8 subsets of a set whose cardinality is 3, since 8 = 23.In the 16th century, Adriaan van Roomen solved the problem using intersecting hyperbolas, but this solution does not use only straightedge and compass constructions. François Viète found such a solution by exploiting limiting cases: any of the three given circles can be shrunk to zero radius (a point) or expanded to infinite radius (a line). Viète's approach, which uses simpler limiting cases to solve more complicated ones, is considered a plausible reconstruction of Apollonius' method. The method of van Roomen was simplified by Isaac Newton, who showed that Apollonius' problem is equivalent to finding a position from the differences of its distances to three known points. This has applications in navigation and positioning systems such as LORAN.Later mathematicians introduced algebraic methods, which transform a geometric problem into algebraic equations. These methods were simplified by exploiting symmetries inherent in the problem of Apollonius: for instance solution circles generically occur in pairs, with one solution enclosing the given circles that the other excludes (Figure 2). Joseph Diaz Gergonne used this symmetry to provide an elegant straightedge and compass solution, while other mathematicians used geometrical transformations such as reflection in a circle to simplify the configuration of the given circles. These developments provide a geometrical setting for algebraic methods (using Lie sphere geometry) and a classification of solutions according to 33 essentially different configurations of the given circles.Apollonius' problem has stimulated much further work. Generalizations to three dimensions—constructing a sphere tangent to four given spheres—and beyond have been studied. The configuration of three mutually tangent circles has received particular attention. René Descartes gave a formula relating the radii of the solution circles and the given circles, now known as Descartes' theorem. Solving Apollonius' problem iteratively in this case leads to the Apollonian gasket, which is one of the earliest fractals to be described in print, and is important in number theory via Ford circles and the Hardy–Littlewood circle method.".
- Problem_of_Apollonius thumbnail Apollonius_problem_typical_solution.svg?width=300.
- Problem_of_Apollonius wikiPageExternalLink Gisch%20and%20Ribando.pdf.
- Problem_of_Apollonius wikiPageID "2642185".
- Problem_of_Apollonius wikiPageRevisionID "603890834".
- Problem_of_Apollonius hasPhotoCollection Problem_of_Apollonius.
- Problem_of_Apollonius subject Category:Conformal_geometry.
- Problem_of_Apollonius subject Category:Euclidean_plane_geometry.
- Problem_of_Apollonius subject Category:History_of_geometry.
- Problem_of_Apollonius subject Category:Incidence_geometry.
- Problem_of_Apollonius comment "In Euclidean plane geometry, Apollonius's problem is to construct circles that are tangent to three given circles in a plane (Figure 1). Apollonius of Perga (ca. 262 BC – ca. 190 BC) posed and solved this famous problem in his work Ἐπαφαί (Epaphaí, "Tangencies"); this work has been lost, but a 4th-century report of his results by Pappus of Alexandria has survived.".
- Problem_of_Apollonius label "Apollonisches Problem".
- Problem_of_Apollonius label "Problem Apoloniusza".
- Problem_of_Apollonius label "Problem of Apollonius".
- Problem_of_Apollonius label "Problema de Apolonio".
- Problem_of_Apollonius label "Problema de Apolónio".
- Problem_of_Apollonius label "Problema di Apollonio".
- Problem_of_Apollonius label "Problème des contacts".
- Problem_of_Apollonius label "Raakprobleem van Apollonius".
- Problem_of_Apollonius label "Задача Аполлония".
- Problem_of_Apollonius label "مسألة أبولونيوس".
- Problem_of_Apollonius label "阿波罗尼奥斯问题".
- Problem_of_Apollonius sameAs Apolloniova_úloha.
- Problem_of_Apollonius sameAs Apollonisches_Problem.
- Problem_of_Apollonius sameAs Απολλώνιο_πρόβλημα.
- Problem_of_Apollonius sameAs Problema_de_Apolonio.
- Problem_of_Apollonius sameAs Problème_des_contacts.
- Problem_of_Apollonius sameAs Problema_di_Apollonio.
- Problem_of_Apollonius sameAs 아폴로니오스의_문제.
- Problem_of_Apollonius sameAs Raakprobleem_van_Apollonius.
- Problem_of_Apollonius sameAs Problem_Apoloniusza.
- Problem_of_Apollonius sameAs Problema_de_Apolónio.
- Problem_of_Apollonius sameAs m.07ts3h.
- Problem_of_Apollonius sameAs Q619449.
- Problem_of_Apollonius sameAs Q619449.
- Problem_of_Apollonius wasDerivedFrom Problem_of_Apollonius?oldid=603890834.
- Problem_of_Apollonius depiction Apollonius_problem_typical_solution.svg.
- Problem_of_Apollonius isPrimaryTopicOf Problem_of_Apollonius.
