Wikipedia:Today's featured article/November 22, 2008

The Problem of Apollonius is a challenge in Euclidean plane geometry to construct circles that are tangent to three given circles in a plane. Apollonius of Perga posed and solved this famous problem in his work Ἐπαφαί ("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 and each solution circle encloses or excludes the three given circles in a different way. 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 to navigation and positioning systems such as GPS. Later mathematicians introduced algebraic methods, which transform a geometric problem into algebraic equations. (more...)

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