# The Problem of Apollonius

What is referred nowadays as the *Problem of Apollonius* is the subject of two lost books *The Tangencies* by Apollonius of Perga (c. 262-190 BC). (This is the same Apollonius who named ellipse, parabola, and hyperbola in his surviving work *Conics*.) We know of the Problem of Apollonius though the multi volume *Collection* of Pappus of Alexandria (c. 290-350) - a famous geometer in his own right. I quote from the fragment in [Greek Mathematical Works, 342-343]:

Next in order are the two books **On Tangencies**. Their enunciations are more numerous, but we may bring these also under one enunciation thus stated:

**
Given three entities, of which any one may be a point or a straight line or a circle, to draw a circle which shall pass through each of the given points, so far as it is points which are given, or to touch each of the given lines.
**

In this problem, according to the number of like or unlike entities in the hypotheses, there are bound to be, when the problem is subdivided, ten enunciations. For the number of different ways in which three entities can be taken out of three unlike sets is ten.

Here is the list of the ten subproblems, with a reasonably common set of notations:

- Three points (PPP)
- Three lines (LLL)
- Two points and one line (PPL)
- One point and two lines (PLL)
- Two points and one circle (PPC)
- One point and two circles (PCC)
- Two lines and one circle (LLC)
- One line and two circles (LCC)
- One point, one line, and one circle (PLC)
- Three circles (CCC)

Pappus continues:

Of these, the first two cases are proved in the fourth book of the first *Elements*, for which reason they will not be described; for to describe a circles through three points, not being in a straight line, is the same thing as to circumscribe a given triangle, and to describe a circle to touch three given straight lines, not being parallel but meeting each other, is the same thing as to inscribe a circle in a given triangle; the case where two of the lines being parallel and one meets them is a subdivision of the second problem but is here given first place. The next six problems in order are investigated in the first book, while the remaining two, the case of two given straight lines and a circle and the case of three circles, are the sole subjects of the second book on account of the manifold positions of the circles and straight lines with respect one to another and the need for numerous investigations of the limits of possibility.

There is no indication that the original solution by Apollonius was incomplete, except a mention by [David Darling, 19] who claims that "the two easiest involve three points or three straight lines and were first solved by Euclid. Solutions to the eight other cases, with the exception of the three-circle problem, appeared in *Tangencies*; however, this work was lost." I wonder whether out of the "manifold positions" of the three-circle problem, Apollonius missed one or two. It is likely he did not. The reason I think so is that in the sixteenth century François Viète managed to reduce by a simple device the (CCC) problem to (PCC). The reduction is so straightforward that it is hard to imagine that a mathematician of Apollonius' calibre may have missed it.

Over the centuries, the problem attracted such famous mathematician as Newton, Euler and René Descartes. [Coolidge] refers to another work that listed some seventy references for the nineteenth century alone, among these were Gauss, Augustin Cauchy, Poncelet, Gergonne, Lazare Carnot, Sophus Lie.

In the special case, where the three given circles are tangent to each other, there are besides the three circles themselves, two additional solutions, known as the Soddy circles. Repeated construction of the Soddy circles leads to Apollonian gasket, an appealing fractal.

Following are the links to already available pages of solutions to the Problem of Apollonius. I shall use the notation \(X(Y)\) to denote the circle with center \(X\) through point \(Y,\) or, if the latter is not important, or clear from the context, \((X)\), or, even more sparingly, with the lower case \(x\) for the center. If there are more points on the circle that need to be mentioned, I'll write, say, \(X(YZ)\) to denote the circle with center \(X\) through pointd \(Y\) and \(Z\). Finally, if the center is not known a priori, then \((YZW)\) will denote the circle through the points \(Y\), \(Z\), \(W\), with more points mentioned as needed.

- Three points (PPP)
- Three lines (LLL)
- Two points and one line (PPL)
- One point and two lines (PLL)
- Two points and one circle (PPC)
- One point and two circles (PCC)
- Two lines and one circle (LLC)
- One line and two circles (LCC)
- One point, one line, and one circle (PLC)
- Three circles (CCC)

Most of the solutions to various cases of the Problem of Apollonius involve several basic constructions:

- How to Construct Tangents from a Point to a Circle
- How to Construct Common Tangents to Two Circles I
- How to Construct Common Tangents to Two Circles II
- How to Construct a Radical Axis
- How to Construct an Orthogonal Circle

### References

- J. L. Coolidge,
*A Treatise On the Circle and the Sphere*, AMS - Chelsea Publishing, 1971 - D. Darling,
*The Universal Book of Mathematics*, John Wiley & Sons, 2004 - I. Thomas,
*Greek Mathematical Works*, v2, Harvard University Press, 2005

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