Neuberg Sangaku

Many of sangaku problems have been thought up in the West, sometimes earlier and sometimes later than they have been published in Japan. For researchers into the Japanese mathematics (wasan) of the Edo period this is of course important to identify the instances where precedence lies in the East. A construction of three tangent circles in what's now known as Malfatti's problem after an 1803 article by Gian Francesco Malfatti (1731-1807) has been proposed some 30 years ealier by the great Japanese mathematican Ajima Naonobu.

The applet below presents a problem posed by J. J. B. Neurberg in 1896 (Methesis, p. 193, problem 1078):


Given a ΔABC, let R, r, ra, rb, rc be the radii of the circumcircle (O), the incircle (I) and the excircles (Ia), (Ib) and (Ic). The fourth tangents common to pairs of the excircles (Ia), (Ib) and (Ic) form a triangle A'B'C'. Show that centers of the incircle of ΔA'B'C' and the circumcircle of ΔIaIbIc coincide and that the radius r' of the former satisfies:

  r' = 2R + r = (r + ra + rb + rc)/2.

The result 2r' = r + ra + rb + rc is known to have been presented in 1803 by Yamamoto Norihisa on a sangaku at the Echigo Hukasan shrine, Niigata prefecture. The tablet has been lost but was recorded in Nakamura's 1830 manuscript Saishi Shinzan.


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  1. Fukagawa Hidetoshi, Tony Rothman, Sacred Mathematics - Japanese Temple Geometry, Princeton University Press, 2008, pp. 298-299


  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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