Problem of Apollonius/Circles

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Classic Problem

Let there be three circles in the plane.

It is required to draw another circle tangent to each of the three.


Solution

It is generically possible to construct such a circle in $8$ different ways.

Each tangent circle encloses a subset of the three original circles.

From Cardinality of Power Set of Finite Set, there are thus $2^3 = 8$ distinct such tangent circles


Proof


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Source of Name

This entry was named for Apollonius of Perga.


Historical Note

The Problem of Apollonius was originally posed, and apparently solved, by Apollonius of Perga in his work Tangencies.

The cases of $3$ points and $3$ straight lines were given by Euclid in his Elements: Book $\text {IV}$.


The case of the $3$ circles was interesting enough to attract the attention of a number of mathematicians of the $17$th century, including Isaac Newton and François Viète.

It was supposedly solved by Elisabeth of the Palatinate during the course of a series of mathematical discussions with René Descartes sometime around or after $1641$.

While this in itself is a remarkable feat, the fact that she achieved it using Descartes' own methods makes it more impressive, as these are not the easiest tools to use to attack the problem.

However, Descartes appears to have been unfairly and cruelly dismissive of her achievement


Sources

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