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
Επαφαι ("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...)
