Two problems on touching circles - ISISS M. Casagrande (Pieve di Soligo)

Sangaku are Japanese geometrical problems which were placed as offerings at Shinto shrines or Buddhist temples in Japan several centuries ago. The aim of our work is to study two of
these Sangaku regarding different configurations of touching circles and to eventually show their connection. Essentially both problems ask to find a circle given a starting configuration built using other circles. Regarding the first problem we first show how to solve it and then, by infinitely iterating the initial geometrical configuration, we build a binary tree structure for the radii of all the generated circles. Showing that the tree structure for the radii is related to the Stern-Brocot tree (a binary tree expressing all positive rational numbers) we provide a method to find the radius of any circle given the radii of the initial two circles. We are also able to give a closed expression for the total area of a particular collection of circles. Finally, using the properties of the inversion transformation we show how to map the first Sangaku to the second one highlighting the fact that they represent actually the same problem.