The configuration of three mutually tangent circles has received particular attention. Generalizations to three dimensions-constructing a sphere tangent to four given spheres-and beyond have been studied. 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.Īpollonius' problem has stimulated much further work. 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 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). Later mathematicians introduced algebraic methods, which transform a geometric problem into algebraic equations. This has applications in navigation and positioning systems such as LORAN. 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. Viète's approach, which uses simpler limiting cases to solve more complicated ones, is considered a plausible reconstruction of Apollonius' method. 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). In the 16th century, Adriaan van Roomen solved the problem using intersecting hyperbolas, but this solution does not use only straightedge and compass constructions. Three given circles generically have eight different circles that are tangent to them (Figure 2), a pair of solutions for each way to divide the three given circles in two subsets (there are 4 ways to divide a set of cardinality 3 in 2 parts). 190 BC) posed and solved this famous problem in his work Ἐπαφαί ( Epaphaí, "Tangencies") this work has been lost, but a 4th-century AD report of his results by Pappus of Alexandria has survived. In Euclidean plane geometry, Apollonius's problem is to construct circles that are tangent to three given circles in a plane (Figure 1). Figure 2: Four complementary pairs of solutions to Apollonius's problem the given circles are black.
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