Apollonian Circles Theorem

The circle we just constructed is known as a Circle of Apollonius. The family of all such circles obtained for different values of r and fixed points A and B is known as the Apollonian Family of Circles defined by the points A and B. Note that one of these circles (corresponding to r = 1) is actually a straight line. It is often convenient to think of straight lines together with common circles as generalized circles, circles of infinite and finite radii.

Since, for the same A and B, each of the Apollonian circles correponds to a different r, no two Apollonian circles intersect. For 0 < r < 1, the circles are closer to A and surround it. For smaller values of r, they are increasingly close to B. For r > 1, the circles surround B and, as r grows become increasingly close to it. For this reason, points A and B are often considered as circles, now point circles, circles of radius 0.