# In the Wasan Spirit

As Tony Rothman explained in his trend-setting article,

There is a word in Japanese, *wasan*, that is used to refer to native Japanese mathematics. *Wasan* is meant to stand in opposition to *yosan*, or Western mathematics. ... To the extent that it makes sense to credit anyone with the founding of wasan, that honor probably goes to Mori and Yoshida (1598-- 1672). ... *Wasan*, though, was created not so much by a few individuals but by something much larger."

And later,

... by the next century, books were being published that contained typical native Japanese problems: circles within triangles, spheres within pyramids, ellipsoids surrounding spheres. The problems found in these books do not differ in any important way from those found on the tablets, and it is difficult to avoid the conclusion that the peculiar flavor of all wasan problems--including the sangaku--is a direct result of the policy of national seclusion.

I do not share Rothman's attribution of nested figures as "native Japanese" (remember the one on Archimedes' tombstone?) or related to the policy of seclusion. (Why not to ascribe the development of projective geometry by J.-V. Poncelet to his longing for the distant France while in Russian captivity?) I also disagree with Rothman's assessment of sangaku popularity. Nonetheless, there are indeed several sangaku dealing with various configurations of tangent circles. Below I discuss a couple of simple ones that appear as likely candidate for the attention of a budding sangaku devotee. One, from P. Yiu's article where he mentions the book by Fukagawa and Pedoe as a possible source, and the other occurred to me through a memory malfunction when I tried to elicit a recollection of P. Yiu's example. Both problems are easily solved with two applications of the Pythagorean theorem.

### Problem 1

The centers A and B of two circles lie on the other circle. Construct a circle tangent to the line AB, to the circle (A) internally, and to the circle (B) externally.

### Problem 2

A circle is tangent internally to a bigger circle and its diameter. Construct the circle tangent to both and to that diameter and express its radius in terms of the large circle.

### References

- T. Rothman,
__Japanese Temple Geometry__,*Scientific American*, May, 1998 H. Fukagawa, D. Pedoe,

*Japanese Temple Geometry Problems*, The Charles Babbage Research Center, Winnipeg, 1989Write to:

Charles Babbage Research Center

P.O. Box 272, St. Norbert Postal Station

Winnipeg, MB

Canada R3V 1L6- P. Yiu,
__Elegant Geometric Constructions__,*Forum Geometricorum*, 5 (2005), pp. 75-96

## 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 © 1996-2017 Alexander Bogomolny

### Solution to Problem 1

Based on the diagram, let AB = a, AF = x, and OF = r (the radius of the sought circle.) Then in ΔBFO,

(a + r)^{2} = r^{2} + (a + x)^{2}.

And in ΔAFO,

(a - r)^{2} = r^{2} + x^{2}.

Subtraction gives

4ar = a^{2} + 2ax,

or

x + a/2 = 2r,

which means that side EF of the square MCEF, M being the midpoint of AB, is a diameter of the sought circle. The construction is now easy.

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Copyright © 1996-2017 Alexander Bogomolny

### Solution to Problem 2

Based on the diagram below, let R be the radius of the middle circle (E), r the unknown radius of the circle in question.

In ΔADE,

(R + r)^{2} = AD^{2} + (R - r)^{2},

or

4Rr = AD^{2}.

In ΔABC,

(2R - r)^{2} = BC^{2} + r^{2},

or

R^{2} - 4Rr = BC^{2}.

Since AD = BC, we have

4Rr = 4R^{2} - 4Rr,

or R = 2r. Center A of the circle can then be found at the intersection of circles with radii 3R/2 centered at E and C.

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