The line segment between these two intersection points is the diameter of the circle of Apollonius.

Hence, line is the common radical axis for each of the three pairs of circles of Apollonius.

The perpendicular bisector of line segment is the Lemoine line, which contains the three centers of the circles of Apollonius.

As well as being the intersections of the circles of Apollonius, each isodynamic point is the intersection points of another triple of circles.

For other circles associated with Apollonius of Perga, please see the disambiguation page, circles of Apollonius.

The circle of Apollonius is any of several types of circles associated with Apollonius of Perga, a renowned Greek geometer.

Stated another way, P is a point on the circle of Apollonius if and only if the cross-ratio [A,B;C,P] is on the unit circle in the complex plane.

A circle can also be defined as the circle of Apollonius, in terms of two different foci, as the set of points having a fixed ratio of distances to the two foci.

Descartes completed (in 1643) the theorem of the three circles of Apollonius, leading to a quadratic equation in 87 terms, each of which is a product of six factors (which, with this method, makes the actual construction humanly impossible).