Geometric puzzles at infinity

Sangaku are an inexhaustible source of geometric wonder. Their figures and the metric relationships they reveal have a particularly strong aesthetic appeal. Some have become classics of geometry. One example is the drawing on a tablet dating from 1820, displayed in Miyagi Prefecture (Honshu), which shows two circles tangent to each other and to the same line. The challenge is to express the distance between the circles' points of tangency with the line in terms of their radii, r and R.

The solution follows directly from applying the Pythagorean theorem to triangle O₁HO₂. We have AB² = O₁O₂² – O₂H, and since O₁O₂ = r + R and O₂H = R – r, a straightforward calculation ultimately gives $\text{AB}=2\sqrt{r\text{R}}\mathrm{.}\backslash text\{AB\}\; =\; 2\; \backslash sqrt\{r\; \backslash text\{R\}\}.$

The ultimate classic ---------------------------

Another figure related to the previous one is emblematic of the Wasan period (17^th century). It appears on a tablet dating from 1824, displayed in Gunma Prefecture (Honshu), a mountainous region north of Tokyo. Take the two circles above and insert a small circle tangent to both larger circles and to the line. What elegant relationship links the radii of the three circles?