The sketch on the right shows some other possibilities. Proof of Apollonius Theorem Notice that the theorem, shows that the set of all apollonian curves consists of the set of circles and lines on the plane. Each family contains one line, so we are associating each line with a family of circles. We shall return to this presently. The second part of the Theorem shows that the points A and B are related to the circle in a similar way.
Apollonian families have many nice properties.