Many a sangaku come in clearly interrelated sequences, although their creation might have been spread of a few decades. We'll start with an example from [Fukagawa and Pedoe, Example 1.6].

A chord AB divides a circle O(r) into two segments. A chain of three contact circles O₃(r₁), O₂(r₂), O₁(r₁) all touch AB and also O(r) internally, as shown in the figure. The tangent at the point of contact of the circle O₂(r₂) and O₁(r₁) meets the circle O(r) in P and Q. Show that Q is the midpoint of the arc AB, and find the length of PQ in terms r₁ and r₂.

Solution

References

1. H. Fukagawa, D. Pedoe, Japanese Temple Geometry Problems, The Charles Babbage Research Center, Winnipeg, 1989

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Copyright © 1996-2008 Alexander Bogomolny

We have two show that the common tangent of the two circles O₂(r₂) and O₁(r₁) passes through the midpoint of the arc AB and also find the length of that tangent inside the circle O(r). The first task has been accomplished elsewhere by noting that the tangent serves as the radical axis of the two circles. Below we prove this result by computing the lengths of several line segments and using the converse of the Pythagorean theorem to show that ΔQTO₂ is right. By the same means we find the length of PQ.

With the reference to the diagram above, there are several right triangles to which we may fruitfully apply the Pythagorean theorem:

from the first we obtain (HO₁)² = 4r₁r₂ as in a related sangaku.

Substituting this into the second equation we get

Further, the similarity triangles HO₁O₂ and GTO₂

From the same triangles, GO₂ / HO₂ = O₂T / O₁O₂ giving

Therefore,

One more application of the Pythagorean theorem helps determine QT from

As the last step, we would like to show that ΔQO₂T is right (which would imply that QT is indeed tangent to the circles O₁(r₁) and O₂(r₂)). To this end, we are in a position to apply the converse of the Pythagorean theorem: suffice it to show that

which is left as an exercise.

From the orthogonality of QT and O₂T we see that triangles QO₂T and QSP are similar (QPS = 90° as subtended by the diameter QS.) Thus we get an additional proportion:

from which

The next two problems are correspondingly [Fukagawa and Pedoe, #1.6.1] and [Fukagawa and Pedoe, #1.6.2]. Both tablets are lost. The first was written in 1857, Ibaragi prefecture, the second in 1806, Nagano prefecture.

The circles are tangent as shown. AB is the diameter of O(r). Find r₁, r₂ and r' in terms of r.

Solution: this is a combination of the above and another sangaku.

AB is no longer a diameter; the problem is to express r₁, r₂ and r' in terms of r and AB.

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