# 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 high-degree equation, one of one thousand and twenty-four 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 *q* = *m*/*a* + *r*/*m* + *d*/*r*.*a*, *m*, *d*, and *r* in terms of *p* and *q*.

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

- Sangaku: Reflections on the Phenomenon
- Critique of My View and a Response
- 1 + 27 = 12 + 16 Sangaku
- 3-4-5 Triangle by a Kid
- 7 = 2 + 5 Sangaku
- A 49
^{th}Degree Challenge - A Geometric Mean Sangaku
- A Hard but Important Sangaku
- A Restored Sangaku Problem
- A Sangaku: Two Unrelated Circles
- A Sangaku by a Teen
- A Sangaku Follow-Up on an Archimedes' Lemma
- A Sangaku with an Egyptian Attachment
- A Sangaku with Many Circles and Some
- A Sushi Morsel
- An Old Japanese Theorem
- Archimedes Twins in the Edo Period
- Arithmetic Mean Sangaku
- Bottema Shatters Japan's Seclusion
- Chain of Circles on a Chord
- Circles and Semicircles in Rectangle
- Circles in a Circular Segment
- Circles Lined on the Legs of a Right Triangle
- Equal Incircles Theorem
- Equilateral Triangle, Straight Line and Tangent Circles
- Equilateral Triangles and Incircles in a Square
- Five Incircles in a Square
- Four Hinged Squares
- Four Incircles in Equilateral Triangle
- Gion Shrine Problem
- Harmonic Mean Sangaku
- Heron's Problem
- In the Wasan Spirit
- Incenters in Cyclic Quadrilateral
- Japanese Art and Mathematics
- Malfatti's Problem
- Maximal Properties of the Pythagorean Relation
- Neuberg Sangaku
- Out of Pentagon Sangaku
- Peacock Tail Sangaku
- Pentagon Proportions Sangaku
- Proportions in Square
- Pythagoras and Vecten Break Japan's Isolation
- Radius of a Circle by Paper Folding
- Review of Sacred Mathematics
- Sangaku à la V. Thebault
- Sangaku and The Egyptian Triangle
- Sangaku in a Square
- Sangaku Iterations, Is it Wasan?
- Sangaku with 8 Circles
- Sangaku with Angle between a Tangent and a Chord
- Sangaku with Quadratic Optimization
- Sangaku with Three Mixtilinear Circles
- Sangaku with Versines
- Sangakus with a Mixtilinear Circle
- Sequences of Touching Circles
- Square and Circle in a Gothic Cupola
- Steiner's Sangaku
- Tangent Circles and an Isosceles Triangle
- The Squinting Eyes Theorem
- Three Incircles In a Right Triangle
- Three Squares and Two Ellipses
- Three Tangent Circles Sangaku
- Triangles, Squares and Areas from Temple Geometry
- Two Arbelos, Two Chains
- Two Circles in an Angle
- Two Sangaku with Equal Incircles
- Another Sangaku in Square
- Sangaku via Peru
- FJG Capitan's Sangaku

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