Configurations Arising from the Three Circle Theorem

Abstract

Problems involving tangent circles have been investigated in both the West and the East. Japan such problems were of special interest in the period. ( Wasan refers to Japanese mathematics developed independently of Western science in the 17th-19th centuries.) researchers often wrote their problems and solutions on a framed board, which was dedicated to a shrine or a temple to express gratitude to the gods. Most such problems were geometric, and the figures were beautifully drawn in color. The board is called a sangaku (votive tablet on mathematics).* It was also a means to publish a discovery or to propose a problem. So mathematicians thereby left many geometric results (in particular on tangent circles or spheres). We will refer to two such results (for extensive references see [5] or [7]). The author discovered an interesting configuration and consequence of this theorem while considering a problem of the mathematician Ushijima [8]: In FIGURE 1, C1, C2, C3, C4 are in circles of the triangles ABD, ADC, AD'C', AB'D' respectively. Given the diameters of C2, C3, C4, find the diameter of C1. The solution essentially states that