Tangent circles

Tangent circles

In geometry, tangent circles (also known as kissing circles) are circles 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.

External/internal tangency; chains of tangent circles

One given circle

Two given circles

teiner 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 interstital 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 the largest three cylinders from a triangular block of marble.

ix 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.

even circles theorem

A chain of six tangent circles is given, each of which is tangent to a seventh given circle. The tangent points of the chain circles with the seventh circle are connected pairwise between opposite circles in the chain, i.e., between circles 1 and 4, 2 and 5 and 3 and 6. These lies are concurrent, i.e., they intersect in the same point.

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

Bibliography

See also

* Tangent lines to circles

External links

*


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