Many a Sangaku come in groups of related problems of which the ones below serve an example. All are easily resolved on the basis of another one from Miyagai prefecture. (So I just give a location and a date. The notations are reasonably obvious.)

• Tokyo prefecture, 1789.

  Answer: r_n = ((n - 2)/√r₁ + 1/√r₂)^-2, n > 2.

• Osaka prefecture, 1801.

  Answer: r₁ = a·(3 - 2√2). r_n = a / (√2 + n)², n > 1.

• Mie prefecture, 1844.

  Answer: r₁ = a/4. r_n = a / (n + 1)², n > 1.

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
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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 49^th Degree Challenge
7. A Geometric Mean Sangaku
8. A Hard but Important Sangaku
9. A Restored Sangaku Problem
   • A better solution to a difficult 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. An Old Japanese Theorem
16. Archimedes Twins in the Edo Period
17. Arithmetic Mean Sangaku
18. Bottema Shatters Japan's Seclusion
19. Circles and Semicircles in Rectangle
20. Circles in a Circular Segment
21. Circles Lined on the Legs of a Right Triangle
22. Equal Incircles Theorem
23. Equilateral Triangle, Straight Line and Tangent Circles
24. Equilateral Triangles and Incircles in a Square
25. Five Incircles in a Square
26. Four Hinged Squares
27. Four Incircles in Equilateral Triangle
28. Gion Shrine Problem
29. Harmonic Mean Sangaku
30. Heron's Problem
31. In the Wasan Spirit
32. Incenters in Cyclic Quadrilateral
33. Japanese Art and Mathematics
34. Malfatti's Problem
35. Maximal Properties of the Pythagorean Relation
36. Neuberg Sangaku
37. Out of Pentagon Sangaku
38. Peacock Tail Sangaku
39. Pentagon Proportions Sangaku
40. Pythagoras and Vecten Break Japan's Isolation
41. Radius of a Circle by Paper Folding
42. Review of Sacred Mathematics
43. Sangaku à la V. Thebault
44. Sangaku and The Egyptian Triangle
45. Sangaku in a Square
46. Sangaku Iterations, Is it Wasan?
47. Sangaku with 8 Circles
48. Sangaku with Three Mixtilinear Circles
49. Sangaku with Versines
50. Sangakus with a Mixtilinear Circle
51. Sequences of Touching Circles
52. Square and Circle in a Gothic Cupola
53. Tangent Circles and an Isosceles Triangle
54. The Squinting Eyes Theorem
55. Steiner's Sangaku
56. Three Incircles In a Right Triangle
57. Three Squares and Two Ellipses
58. Three Tangent Circles Sangaku
59. Triangles, Squares and Areas from Temple Geometry
60. Two Arbelos, Two Chains
61. Two Circles in an Angle
62. Two Sangaku with Equal Incircles

Copyright © 1996-2009 Alexander Bogomolny