A Sangaku with Many Circles and Some: What Is This About?
A Mathematical Droodle

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Solution

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Copyright © 1996-2018 Alexander Bogomolny

The applet purports to suggest the following 1837 sangaku [Temple Geometry, 1.5.7] from the Aichi prefecture:

AB is a diameter of O(r) and forms two semicircles. In one of these the equal circles O1(r1) and O'1(r1) touch AB and also O(r) internally. The circles O2(r1) and O'2(r1) are mirror images of these two circles in AB. The circle O(r3) touches the 4 circles we have just described externally, and the circle O4(r4) touches O1(r1) and O'1(r1) externally and O(r) internally. Show that   r3 = r4. In the particular case where r3 = r/9, show that   r3 = r4 = r5, where O5(r5) has interior contact, as shown, with both O1(r1) and O'1(r1).

We'll proceed a step at a time. First, let X be the points of contact of O₁(r₁) with AB and x = OX.

By the Pythagorean theorem,

(r - r1)² = x² + (r1)².

From which

(1)	r1 = (r² - x²) / 2r.

Next we find r₃:

(r3 + r1)² = x² + (r1)².

implying a quadratic equation

(r3)² + 2r1r3 - x² = 0.

The equation has two real roots, a negative one and a positive one, the latter being

r3 = - r1 + √(r1² + x²) = 0.

Substituting r₁ from (1) gives

(2)	r3 = x² / r.

Next in line is r₄:

for which the Pythagorean theorem provides

(r - r1 - r4)² + x² = (r1 + r4)².

One simplification is immediate:

r² + x² = 2r(r1 + r4),

Substituting r₁ from (1) gives

r² + x² = r² - x² + 2r·r4,

from which

(2)	r4 = x² / r.

Thus we see that indeed r₃ = r₄.

When x = r/3, r₃ = r₄ = r/9. Note that, for this value of x, r₁ = 4r/9. Also, circles O₁(r₁) and O'₁(r₁) have centers 2x = 2r/3 apart giving the width of the lens shaped intersection as

2r1 - 2x = 8r/9 - 2r/3 = 2r/9,

implying r₅ = r/9 = r₃ = r₄, for x = r/3.

The sangaku ends here, but having a dynamic applet invites an investigation. If the three equal circles grow while the quadruplets become small, the former eventually form 5 regions, into which one may want to inscribed circles:

When the five circles are equal, their common radius is r/5; obviously. More can be said, viz., the outer four circles are always equal. Furthermore, their radius is exactly r₁ increasing the number of equal circles to 8. The proof is not difficult and is left as an exercise.

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

2. H. Fukagawa, A. Rothman, Sacred Geometry: Japanese Temple Geometry, Princeton University Press, 2008, p. 101

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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
   • A Simple Solution to a Difficult Sangaku Problem
   • A Trigonometric 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. 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
    • Equal Incircle Theorem, Angela Drei's Proof
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
    • Four Incircles in an Equilateral Triangle, a Sangaku
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
    • An Extension of a Sangaku with Touching Circles
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