Sangaku with Three Mixtilinear Circles

Sangaku traditionally contained a diagram and a question concerning the diagram. Sometimes there were also instructions for constructing of the depicted objects and occasionally a solution to the problem. One such sangaku with a solution could be found in [Smith and Mikami, p. 185]:

There is a circle in which a triangle and three circles, A, B, C, are inscribed in the manner shown in the figure. Given the diameters of the three inscribed circles, required the diameter of the circumscribed circle.

(The three inscribed circles are the mixtilinear circles in the inscribed triangle. Each mixtilinear circle is inscribed in an angle and touches the circumcircle of the triangle. In the applet below, I labeled A, B, C the vertices of the triangle with understanding that the circles are also labeled A, B, C depending on the angle into which each is inscribed.)

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at, download and install Java VM and enjoy the applet.

What if applet does not run?

The sangaku comes with a solution that is representative of the contemporary mathematical style:

Let the respective diameters be x, y, and z, and let xy = a. Then from a² take [(x - y)z]². Divide a by the remainder and call the result b. Then from (x + y)z take a and divide 0.5 by this remainder and add b, and then multiply by z and by a. The result is the diameter of the circumscribed circle.

Smith and Mikami note that to this rule is appended, with some note of pride, the words: "Feudal District of Kakegawa in Yenshu Province, third month of 1795, Miyajima Sonobei Keichi, pupil of Fujita Sadasuke of the School of Seki."

The rule can be translated as

xyz [xy / (x²y² - (x - yx²) + 0.5 /((x + y)z - xy)]

The applet let's you verify that the rule does work.


  1. D. E. Smith and Yoshio Mikami, A History of Japanese Mathematics, Dover, 2004 (originally 1914)

Related material

Mixtilinear Incircles

  • Mixtilinear Circles and Concurrence
  • Sangakus with a Mixtilinear Circle
  • Radius and Construction of a Mixtilinear Circle
  • Construction and Properties of Mixtilinear Incircles
  • Construction and Properties of Mixtilinear Incircles 2
  • Conic in Mixtilinear Incircles
  • A Sangaku: Two Unrelated Circles
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    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
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    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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