Apollonius Problem: What is it?
A Mathematical Droodle

The problem was posed and solved by one of the greatest Greek geometers, Apollonius of Perga (ca. 260-170 B.C.) However, his original solution that was included in his treatise De Tactionibus has been lost. Nowadays, a multitude of solutions is available with contributions from some very famous mathematicians, like François Viète, C. F. Gauss, J. D. Gergonne.

Tradionally, the problem covers several special cases. For example, one or more of the given circles may degenerate into a point. (If all 3 do, the problem is then reduced to constructing the circumcircle of a given triangle. Note that if the three points are collinear, the circumcircle degenerates into a straight line. If the given circles touch at the same point, there is an infinitude of solutions.

Quite often, especially in Inversive Geometry, straight lines are also considered circles - circles with infinite radius and center at infinity. In particular, if the three sides lines of a triangle are looked at as such circles of infinite radius, the incircle and the three excircles of the triangle are solutions to the Apollonius' problem. The applet above eschews those cases. However, even if we only concentrate on the circles proper, the problem may not have a solution. This happens, for example, if the given circles are concentric. But there also could be an infinite number of solutions if the three circles all touch at a point. (Thanks to Wilson Stothers for pointing out this possibility.)

As usual in Greek geometry, construction of the tangent circles must be executed with the customary means: straightedge and compass.

If two circles touch, the point of tangency lies on the line connecting their centers. The circles may touch externally or internally (when one of the circles lies in the interior of the other.) If they touch externally, the distance between their centers equals the sum of their radii. If they touch internally, that distance equals the difference of the radii. We may use Analytic Geometry to express the conditions of tangency of the sought circle and the three given ones.

Let given circles have radii r₁, r₂, and r₃, and centers at (x₁, y₁), (x₂, y₂), and (x₃, y₃). We are looking for a circle of (unknown) radius r centered at (an unknown point) (x, y). The conditions of tangency result in three quadratic equations:

The signs in the three equations can be chosen independently of each other. Therefore the problem may have as many as 2·2·2 = 8 solutions, but does not always have that many. In a general case, subtracting one of the equations from the remaining two results in a system of 2 linear equations, from which x and y may be expressed in terms of r. Substitution then leads to a quadratic equation in r which implies that the unknown quantities x, y, r can be expressed in terms of quadratic radicals as functions of the given centers and radii. As we know, this means that solutions to the Apollonius problem are constructible with straightedge and compass.

Wilson Stothers from the University of Glasgow, UK, went into a much greater detail describing all possible relative configurations of the circles that affect the total number of solutions to the Apollonius' problem.