Peacock's Tail Sangaku
Many a sangaku deal with inscribed circles. While mostly elementary and solved easily by a few applications of Pythagorean theorem they were probably more a work of art than mathematics. Here's one such (to my taste) example. The sangaku was hung by a woman, Okuda Tsume, in the year 1865 at the Meiseirinji temple in Ogaki City, Gifu prefecture.
In a circle of diameter 2R, draw two tangent arcs of radius R, and then ten inscribed circles, two of diameter R; four red of radius t and four blue of radius t'. Show that
References
 H. Fukagawa, A. Rothman, Sacred Mathematics: Japanese Temple Geometry, Princeton University Press, 2008, p. 107
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Copyright © 19962018 Alexander Bogomolny
In a circle of diameter 2R, draw two tangent arcs of radius R, and then ten inscribed circles, two of diameter R; four red of radius t and four blue of radius t'. Show that
Solution
The solution is based on the following diagram
For t we consider the left dashed triangle:
(R/2)² + (R/2 + t)² = (R  t)².
For t', consider the two right dashed triangles with a common altitude. Applying the Pythagorean theorem twice and equating the altitudes gives:
(R  t')²  (t')² = (R + t')²  (R  t')².
The two equations are easily manipulated into the desired identities.
Sangaku

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Copyright © 19962018 Alexander Bogomolny