The apollonius metric theorem
The metric solution of the apollonius problem is quite complex, and it has been considered from many viewpoints. We discuss
this nearer elsewhere, and brings here the formula of the radius of an apollonius circle given
the inverse radiuses of the three other circles, and the distangles between them.
There are eight solutions to the apollonius problem in the general case, and
the above formula gives two of them. When we add the two solutions, the square root expression is falling, and we get
as sum a fraction of two determinants. This sum is a linear relation. The other three pairs of apollonius circles gives
rise to similar formulas, and a combination of all four, gives a simple relation among all eight inverse radiuses.
There must be taken consideration of how the sign of the radiuses are. We say that the tangency between two circles is
negative if they touch each other outwardly, and positive if the one is inside the other. The sign of an apollonius circle is
defined as the product of the three tangencies. Then we have the relation.
Theorem: Taking into account the sign of the radiuses of the
apollonius circles, the sum of the eight inverse radiuses of the is zero.
The theorem can be tried here: Eight apollonius circles.
From this theorem several can be deduced, and some of the species are developed here:
