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geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, tangent circles (also known as kissing circles) are circles in a common plane that intersect in a single point. There are two types of tangency: internal and external. Many problems and constructions in geometry are related to tangent circles; such problems often have real-life applications such as
trilateration Trilateration is the use of distances (or "ranges") for determining the unknown position coordinates of a point of interest, often around Earth ( geopositioning). When more than three distances are involved, it may be called multilateration, for ...
and maximizing the use of materials.


Two given circles

Two circles are mutually and externally tangent if distance between their centers is equal to the sum of their radiiWeisstein, Eric W. "Tangent Circles." From MathWorld--A Wolfram Web Resource
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Steiner chains


Pappus chains


Three given circles: Apollonius' problem

Apollonius' problem is to construct circles that are tangent to three given circles.


Apollonian gasket

If a circle is iteratively inscribed into the interstitial curved triangles between three mutually tangent circles, an Apollonian gasket results, one of the earliest fractals described in print.


Malfatti's problem

Malfatti's problem is to carve three cylinders from a triangular block of marble, using as much of the marble as possible. In 1803, Gian Francesco Malfatti conjectured that the solution would be obtained by inscribing three mutually tangent circles into the triangle (a problem that had previously been considered by Japanese mathematician Ajima Naonobu); these circles are now known as the Malfatti circles, although the conjecture has been proven to be false.


Six circles theorem

A chain of six circles can be drawn such that each circle is tangent to two sides of a given triangle and also to the preceding circle in the chain. The chain closes; the sixth circle is always tangent to the first circle.


Generalizations

Problems involving tangent circles are often generalized to spheres. For example, the Fermat problem of finding sphere(s) tangent to four given spheres is a generalization of
Apollonius' problem 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 (c. 262 190 BC) posed and solved this famous problem in his work (', "Tangenci ...
, whereas Soddy's hexlet is a generalization of a Steiner chain.


See also

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Tangent lines to circles In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
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Circle packing theorem The circle packing theorem (also known as the Koebe–Andreev–Thurston theorem) describes the possible tangency relations between circles in the plane whose interiors are disjoint. A circle packing is a connected collection of circles (in gen ...
, the result that every planar graph may be realized by a system of tangent circles * Hexafoil, the shape formed by a ring of six tangent circles * Feuerbach's theorem on the tangency of the nine-point circle of a triangle with its incircle and excircles * Descartes' theorem * Ford circle * Bankoff circle * Archimedes' twin circles * Archimedean circle * Schoch circles * Woo circles * Arbelos *
Ring lemma In the geometry of circle packings in the Euclidean plane, the ring lemma gives a lower bound on the sizes of adjacent circles in a circle packing. Statement The lemma states: Let n be any integer greater than or equal to three. Suppose that the ...


References


External links

* {{MathWorld, title=Tangent circles, urlname=TangentCircles Circles