Sangakus with a Mixtilinear Circle
What Is This About?
A Mathematical Droodle

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Copyright © 1996-2018 Alexander Bogomolny

The applet purports to suggest the following sangaku [Temple Geometry, 2.3.3]:

Triangle ABC is inscribed in the circle O(R) and AB is a diameter. The circle O1(r1) touches CA and CB and touches O(R) internally, and I(r) is the incircle of triangle ABC. Show that

r1 = 2r.

(This is a 1842 Sangaku from the Iwate prefecture. Another sangaku (2.2.7) does not mention the incircle but requests a proof of r1 = a + b - c, where a, b are the legs and c the hypotenuse of ΔABC. This one was written in 1893, in the Fukusima prefecture.)

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 https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

What if applet does not run?

The formulation of the second sangaku suggests there is a direct way to calculate r1. I do not see it at this point. The problem is solved with a reference to a more general case:

r = r1 cos2(α/2).

Since in the present problems α = 90°, cos(α/2) = √2/2 and, as a consequence, r = r1/2, as required.

References

  1. 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

Related material
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Mixtilinear Incircles

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

    • Sangaku

    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

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