Problem of Apollonius

In <a href="/facts/Euclidean_geometry/V6cTcoCy">Euclidean plane geometry</a>, Apollonius's problem is to construct circles that are <a href="/facts/Tangent/sF0jH3EI">tangent</a> to three given circles in a plane (Figure 1). <a href="/facts/Apollonius_of_Perga/DgpSjkcW">Apollonius of Perga</a> (c. 262 BC – c. 190 BC) posed and solved this famous problem in his work Ἐπαφαί (Epaphaí, "Tangencies"); this work has been <a href="/facts/Lost_literary_work/Hm0hsUIP">lost</a>, but a 4th-century AD report of his results by <a href="/facts/Pappus_of_Alexandria/UABLMu0m">Pappus of Alexandria</a> has survived. 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 <a href="/facts/Cardinality/m6VPojwT">cardinality</a> 3 in 2 parts).
In the 16th century, <a href="/facts/Adriaan_van_Roomen/MEGYUpU8">Adriaan van Roomen</a> solved the problem using intersecting <a href="/facts/Hyperbola/6R6ER0mY">hyperbolas</a>, but this solution does not use only <a href="/facts/Straightedge_and_compass/lbaHf2m7">straightedge and compass</a> constructions. <a href="/facts/Fran%25C3%25A7ois_Vi%25C3%25A8te/8ebmRTjW">François Viète</a> found such a solution by exploiting <a href="/facts/Limiting_case_(mathematics)/UEehGCom">limiting cases</a>: 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 <a href="/facts/Isaac_Newton/4L7gTncN">Isaac Newton</a>, 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 <a href="/facts/LORAN/raqaI2b9">LORAN</a>.
Later mathematicians introduced algebraic methods, which transform a geometric problem into <a href="/facts/Algebraic_equation/1QauIGxs">algebraic equations</a>. These methods were simplified by exploiting <a href="/facts/Symmetry_in_mathematics/KtLgoqyX">symmetries</a> 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). <a href="/facts/Joseph_Diaz_Gergonne/B8ggHgVo">Joseph Diaz Gergonne</a> used this symmetry to provide an elegant straightedge and compass solution, while other mathematicians used <a href="/facts/Transformation_(geometry)/qRjrasJf">geometrical transformations</a> such as <a href="/facts/Inversive_geometry/VqFmbfDI">reflection in a circle</a> to simplify the configuration of the given circles. These developments provide a geometrical setting for algebraic methods (using <a href="/facts/Lie_sphere_geometry/eVU6np2r">Lie sphere geometry</a>) 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 <a href="/facts/Higher_dimension/0ktmUyac">beyond</a> have been studied. The configuration of three mutually tangent circles has received particular attention. <a href="/facts/Ren%25C3%25A9_Descartes/ohqj6DMf">René Descartes</a> gave a formula relating the radii of the solution circles and the given circles, now known as <a href="/facts/Descartes%2527_theorem/QukIe5sE">Descartes' theorem</a>. Solving Apollonius' problem iteratively in this case leads to the <a href="/facts/Apollonian_gasket/4lrnp6zV">Apollonian gasket</a>, which is one of the earliest <a href="/facts/Fractal/sXj38JeN">fractals</a> to be described in print, and is important in <a href="/facts/Number_theory/zhoa7zrc">number theory</a> via <a href="/facts/Ford_circles/kYYWCcgB">Ford circles</a> and the <a href="/facts/Hardy%25E2%2580%2593Littlewood_circle_method/YIY1tt5w">Hardy–Littlewood circle method</a>.

Problem of Apollonius open-in-new

Problem of Apollonius