# A Sangaku by a Teen

It is said that during the Edo period hanging sangaku in temples and shrines was so popular that even women and children partook of the activity. While I am skeptical that sangaku was so popular as to involve a significant slice of the population, some sangaku have indeed been created by children. Here's one example.

In a blue equilateral triangle, three green circles of radius *a*, four red circles of radius *b*, and six white circles of radius *c* touch each other as shown. If *R* is the radius of the outer circle, and *r* is the radius of the dashed circle, find *c* in terms of *r*.

This sangaku has been hung by Tanabe Shigetoshi, aged fifteen, in the year 1865 at the Meiseirinji temple in Ogaki City, Gifu prefecture.

The sangaku has a very simple solution, although it was anything but simple to come up with such a configuration.

### References

- H. Fukagawa, A. Rothman,
*Sacred Mathematics: Japanese Temple Geometry*, Princeton University Press, 2008, p. 97

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Copyright © 1996-2018 Alexander Bogomolny

In a blue equilateral triangle, three green circles of radius *a*, four red circles of radius *b*, and six white circles of radius *c* touch each other as shown. If *R* is the radius of the outer circle, and *r* is the radius of the dashed circle, find *c* in terms of *r*.

### Solution

By direct inspection

r | = 3b + 4c, |

R | = 5b + 4c, |

R | = b + 2a, |

a + b | = 2b + 4c. |

Solving these simultaneously we see that *b* = 2*c*,*a* = 6*c*,*r* = 10*c*.

## 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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Copyright © 1996-2018 Alexander Bogomolny

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