Tangent circles

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In geometry, 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 and maximizing the use of materials.

Two given circles[edit]

Ellipse and hyperbola as the locus of centers of circles tangent to two given intersecting circles.

Steiner chains[edit]

Main article: Steiner chain

Pappus chains[edit]

Main article: Pappus chain

Three given circles: Apollonius' problem[edit]

Main article: Problem of Apollonius

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

Apollonian gasket[edit]

Main article: 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[edit]

Main article: Malfatti circles

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[edit]

Main article: 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.

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