It is often intimated that some of the sangaku - geometric problems carved on colorful wooden tablets - have been posted by clever teenagers. Of course, 200-400 years later, no one can be certain about that, but if there was indeed a problem suitable for an early age, the one below fits the bill (Kiyomizu Temple, Kyoto Prefecture).

Squares and circles are inscribed in successive right triangles. What is the relation between the radii of the circles?

Solution

Copyright © 1996-2008 Alexander Bogomolny

When a square is inscribed in a right triangle with one side on the hypotenuse of the latter, it cuts off three smaller triangles all similar to the initial one. In the problem, the process is repeated with the largest of the three and then again with the largest of the newly created pieces. In all cases, a square is inscribed into similar triangles.

So again, let there be a right triangle with hypotenuse c₁ and c₂ be the hypotenuse of the largest of the three pieces cut off from the triangle by the inscribed square. q = c₁ / c₂. If we repeat the operation with the largest triangle we'll get a smaller triangle of hypotenuse c₃ such that q = c₂ / c₃, because of the similarity.

c₂ appears to be the mean proportional between c₁ and c₃. But then again, because of the similarity, the same is true for any linear element in the triangles, not only the hypotenuse. In particular, the radii of the three circles constructed similarly in the similar triangles satisfy the mean proportional condition:

In other words, the radius of middle (blue) circle is the geometric mean of the radii of the two red circles.

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

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

Copyright © 1996-2008 Alexander Bogomolny