The Apollonian Circle

An Apollonian circle is part of a geometric set where each circle is constructed so that it maintains a fixed ratio of distances to two given points, called foci. Imagine two fixed points, and for any circle in this set, the point on the circle is always proportionally closer to one focus than to the other, based on a constant ratio. This creates a family of circles, all related by this ratio, and they trace intricate patterns known as Apollonian circles. They are fundamental in understanding certain types of geometric transformations and are connected to topics like inversions and circle packings.