Tangent circles

Two given circles

Two circles are mutually and externally tangent if the distance between their centers is equal to the sum of their radii.

Steiner chains

Main article: Steiner chain

Pappus chains

Main article: Pappus chain

Three given circles: Apollonius' problem

Main article: Problem of Apollonius

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

Apollonian gasket

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

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

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.

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.

References

References

1. Weisstein, Eric W.. "Tangent Circles".