Descartes integer formation
One of the most interesting feature of the Descartes theorem is that one can find integer structures associated with it. This connection has to do with the cube numbers, and with the twin circles of the Descartes formation. The twin circles of the Descartes formation is the four circles that are orthogonal to each three of the Descartes circles. It is a remarkable fact that these circles also make an Descartes formation, and the inverse radiuses of these circles are also connected by the same formula. In addition 
Demonstration
