It has been some years that i had gone through the Apollonius' problem. I happened to come across it while browsing through Wikipedia. So here it is :

The problem states that : "Given three objects in the plane, each of which may be a circle C, a point P (a degenerate circle), or a line L (part of a circle with infinite radius), find another circle that is tangent to (just touches) each of the three".

There are ten combinations possible through combinations of point, line and circle. The most difficult case, to find a tangent circle to any three other circles. The resultant solution is here.

These circles are known as Apollonius' Circles. A combination of 8 solutions.

If you observe carefully, the circles in the solution are generated in pairs and are conjugate to one another. Hence forming a symmetry. 

Carrying forward this problem, if the 3 circles in the problem are mutually tangential to one another, then the 8 circles of the solution collapse into 2, as seen here

These circles are known as Soddy Circles.