tangent circles


geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

, 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 In geometry, the tangent line (or simply tangent) to a plane curve at a given Point (geometry), point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitesimal, infinitely cl ...

: internal and external. Many problems and constructions in geometry are related to tangent circles; such problems often have real-life applications such as
trilateration True-range multilateration is a method to determine the location of a movable vehicle or stationary point in space using multiple ranges (distances) between the vehicle/point and multiple spatially-separated known locations (often termed 'statio ...

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

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, 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 , also known as Ajima Manzō Chokuyen, was a Japanese mathematician of the Edo period.Smith, David. (1914). His Dharma name was (祖眞院智算量空居士). Work Ajima is credited with introducing calculus into Japanese mathematics. The sig ...
); these circles are now known as the , 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.


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, whereas Soddy's hexlet is a generalization of a Steiner chain.

See also

* Tangent lines to circles * Circle packing theorem, 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


External links

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