Geometry Problem 1081: Solving Sangaku Japanese Problem: Equilateral Triangle, Inscribed Circle, Inradius, Tangent Circles, Radius, Tangent Line

In the figure below, an equilateral triangle ABC is shown with an inscribed circle of radius r. The figure also includes three yellow circles of radius a, seven green circles of radius b, six white circles of radius c, and circles C₁ that touch each other as depicted. Prove the following equations: \((1)\, a = \dfrac{3}{5}r, \,(2)\, b = \dfrac{r}{5}, \,(3)\, c=\dfrac{r}{10}\).


Reference: Fukagawa Hidetoshi, Tony Rothman, Sacred Mathematics: Japanese Temple Geometry (Princeton University Press, 2008).

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