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.



  1. T. Rothman, Japanese Temple Geometry, Scientific American, May, 1998
  2. H. Fukagawa, D. Pedoe, Japanese Temple Geometry Problems, The Charles Babbage Research Center, Winnipeg, 1989

    Write to:

    Charles Babbage Research Center
    P.O. Box 272, St. Norbert Postal Station
    Winnipeg, MB
    Canada R3V 1L6

  3. P. Yiu, Elegant Geometric Constructions, Forum Geometricorum, 5 (2005), pp. 75-96


  1. Sangaku: Reflections on the Phenomenon
  2. Critique of My View and a Response
  3. 1 + 27 = 12 + 16 Sangaku
  4. 3-4-5 Triangle by a Kid
  5. 7 = 2 + 5 Sangaku
  6. A 49th Degree Challenge
  7. A Geometric Mean Sangaku
  8. A Hard but Important Sangaku
  9. A Restored Sangaku Problem
  10. A Sangaku: Two Unrelated Circles
  11. A Sangaku by a Teen
  12. A Sangaku Follow-Up on an Archimedes' Lemma
  13. A Sangaku with an Egyptian Attachment
  14. A Sangaku with Many Circles and Some
  15. A Sushi Morsel
  16. An Old Japanese Theorem
  17. Archimedes Twins in the Edo Period
  18. Arithmetic Mean Sangaku
  19. Bottema Shatters Japan's Seclusion
  20. Chain of Circles on a Chord
  21. Circles and Semicircles in Rectangle
  22. Circles in a Circular Segment
  23. Circles Lined on the Legs of a Right Triangle
  24. Equal Incircles Theorem
  25. Equilateral Triangle, Straight Line and Tangent Circles
  26. Equilateral Triangles and Incircles in a Square
  27. Five Incircles in a Square
  28. Four Hinged Squares
  29. Four Incircles in Equilateral Triangle
  30. Gion Shrine Problem
  31. Harmonic Mean Sangaku
  32. Heron's Problem
  33. In the Wasan Spirit
  34. Incenters in Cyclic Quadrilateral
  35. Japanese Art and Mathematics
  36. Malfatti's Problem
  37. Maximal Properties of the Pythagorean Relation
  38. Neuberg Sangaku
  39. Out of Pentagon Sangaku
  40. Peacock Tail Sangaku
  41. Pentagon Proportions Sangaku
  42. Proportions in Square
  43. Pythagoras and Vecten Break Japan's Isolation
  44. Radius of a Circle by Paper Folding
  45. Review of Sacred Mathematics
  46. Sangaku à la V. Thebault
  47. Sangaku and The Egyptian Triangle
  48. Sangaku in a Square
  49. Sangaku Iterations, Is it Wasan?
  50. Sangaku with 8 Circles
  51. Sangaku with Angle between a Tangent and a Chord
  52. Sangaku with Quadratic Optimization
  53. Sangaku with Three Mixtilinear Circles
  54. Sangaku with Versines
  55. Sangakus with a Mixtilinear Circle
  56. Sequences of Touching Circles
  57. Square and Circle in a Gothic Cupola
  58. Steiner's Sangaku
  59. Tangent Circles and an Isosceles Triangle
  60. The Squinting Eyes Theorem
  61. Three Incircles In a Right Triangle
  62. Three Squares and Two Ellipses
  63. Three Tangent Circles Sangaku
  64. Triangles, Squares and Areas from Temple Geometry
  65. Two Arbelos, Two Chains
  66. Two Circles in an Angle
  67. Two Sangaku with Equal Incircles
  68. Another Sangaku in Square
  69. Sangaku via Peru
  70. FJG Capitan's Sangaku

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Copyright © 1996-2018 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 = r2 + (a + x)2.

And in ΔAFO,

(a - r)2 = r2 + x2.

Subtraction gives

4ar = a2 + 2ax,


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 © 1996-2018 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.


(R + r)2 = AD2 + (R - r)2,


4Rr = AD2.


(2R - r)2 = BC2 + r2,


R2 - 4Rr = BC2.

Since AD = BC, we have

4Rr = 4R2 - 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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Copyright © 1996-2018 Alexander Bogomolny