Tangent circles

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.

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.

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.

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.