Circles of Apollonius

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Figure 1: The eight solutions to Apollonius' problem, where the three fixed circles are shown in black.

Apollonius of Perga was a renowned Greek geometer whose work often involved circles. For example,

He showed that a circle can be defined as the set of points in a plane that have a specified ratio of distances to two fixed foci. (See Circle for details.)

Using that definition, he identified two families of mutually orthogonal circles. The first family consists of the circles with all possible distance ratios to two fixed foci, whereas the second family consists of all possible circles that pass through the two foci. These circles form the basis of bipolar coordinates and are usually called the Apollonian circles.

He posed and solved Apollonius' problem, which is to identify a circle that is simultaneously tangent to three specified circles (Figure 1). As shown here, there are eight possible solutions. Repeatedly carrying out this construction produces the Apollonian gasket, which has a fractal dimension.

[edit] Technical definition

Circles of Apollonius may be used as a technical term to denote three special circles $\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{C}_{3}$ defined by an arbitrary triangle $A 1 A 2 A 3$ . The circle $\mathcal{C}_{1}$ is defined as the unique circle passing through the triangle vertex $A 1$ that maintains a constant ratio of distances to the other two vertices $A 2$ and $A 3$ (cf. Apollonius' definition of the circle above). Similarly, the circle $\mathcal{C}_{2}$ is defined as the unique circle passing through the triangle vertex $A 2$ that maintains a constant ratio of distances to the other two vertices $A 1$ and $A 3$ , and so on for the circle $\mathcal{C}_{3}$ .

All three circles intersect the circumcircle of the triangle orthogonally. All three circles pass through two points, denoted as the isodynamic points $S$ and $S^{\prime}$ of the triangle. The line connecting these common intersection points is the radical axis for all three circles. The two isodynamic points are inverses of each other relative to the circumcircle of the triangle.

Remarkably, the centers of these three circles fall on a single line (the Lemoine line). This line is perpendicular to the radical axis defined by the isodynamic points $S$ and $S^{\prime}$ .

These "circles of Apollonius" should not be confused with the Apollonian circles, which have a different technical definition.

[edit] An additional meaning?

The following definition has been suggested as a "circle of Apollonius":

The circle that touches all three excircles of a triangle and encompasses them.

This circle can be constructed by solving Apollonius' problem for the three excircles of a given triangle. One of the solutions should be the nine-point circle, which is externally tangent to all three excircles. Intriguingly, the circle described here is internally tangent. A reference for this "circle of Apollonius" and some of its properties would be welcome! :)