Gion Shrine Problem
This is probably the most notorious of the geometric sangaku. The tablet was hung at the Zenkoji temple by Saito Mitsukuni in 1815 [Fukagawa and Rothman, p. 250]. On the tablet Saito wrote:
This problem was first proposed by Tsuda Nobuhisa in 1749 on a sangaku of the Gion Shrine in Kyoto. Tsuda derived the answer with a highdegree equation, one of one thousand and twentyfour degrees. But the famous mathematician Ajima Naonobu showed how to solve it with an equation of only the tenth degree in the variable a. On this tablet, I will show Ajima's equation.
Ajima is on the record to have submitted his solution in 1774 which brought him great fame as a mathematician.
In a circular segment with base AB of length a and altitude m, there are a circle of radius r inscribed in one half of the segment and a square of side d inscribed in the other half, as shown.
Form p = a + m + d + r and
Following Fukagawa and Rothman I, too, admit to having no idea where or how approach that problem.
References
 H. Fukagawa, A. Rothman, Sacred Mathematics: Japanese Temple Geometry, Princeton University Press, 2008
Sangaku

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