Morphomatic

The problem of Apollonius is one of the most famous construction problems in geometry. The task is to construct circles that touch three given circles. It is not known how Apollonius himself did the construction, but in modern time several solutions are given. The fist who accomplished this was Viete, and later Newton also found a solution.

The morphological significance of the problem is that there are ten special cases of the problem. There is the general situation when all tree starting circles are actual circles. Then there are all variants when the circle becomes points or lines, so altogether there are ten variants of the problem.

One of the simplest solution is due to Gergonne; this is a general solution, and it is this that we shall consider. His solution is almost projective in its character, and is quite symmetric. It involves the common line of three circles, the ortosenter, and pol and polare. From the general solution we shall see how this takes special forms, and how far we can go in this different detections.

Gergonnes solution

The Gergonnes solution involves several elements which must be known first. These are the common line of three circles, the ortocenter of three circles, and the concept of pole and polare.