Two Arbelos, Two Chains

Two Arbelos, Two Chains

There is a Pappus' chain in arbelos, the shoemaker's knife, and, vacuously, two of them in two cobbler implements. In a particular sangaku documented as 1.8.5 in the collection by Fukagawa and Pedoe the two devices have much on common which induces a relationship between the circles in their chains:

Points T, A, B, C are collinear and AB = BC = CT = 2r. Circles S₁(3r) and S₂(2r) are drawn on AT and BT respectively as diameters. We consider the chain of contact circles O_i(r_i), i = 1, 2, ..., where O₁(r₁) touches C₁(r), drawn on AB as diameter, touches S₁(3r) internally and S₂(2r) externally, and so on. We also use the circles C₂(r) and C₃(r) with respective diameters BC and CT to construct another chain of contact circles T_i(t_i), i = 1, 2, ..., as the figure makes clear. Prove that

t_n / (t_n / r_n - 3) = 2r / 13.

Note that the required identity is likely to be faulty, for it's quite obvious from the construction that t_n <  r_n, so that the fraction in the denominator is less the one making the denominator negative and, with it, the whole of the left side. Thus the question is to find a relation in the spirit of the required one.

Solution

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Copyright ? 1996-2014 Alexander Bogomolny

Points T, A, B, C are collinear and AB = BC = CT = 2r. Circles S₁(3r) and S₂(2r) are drawn on AT and BT respectively as diameters. We consider the chain of contact circles O_i(r_i), i = 1, 2, ..., where O₁(r₁) touches C₁(r), drawn on AB as diameter, touches S₁(3r) internally and S₂(2r) externally, and so on. We also use the circles C₂(r) and C₃(r) with respective diameters BC and CT to construct another chain of contact circles T_i(t_i), i = 1, 2, ..., as the figure makes clear. Prove that

t_n / (t_n / r_n - 3) = 2r / 13.

This sangaku is said to be written in 1842 in the Nagano prefecture. The table has since disappeared.

Solution

We make use of a useful formula twice getting, for the small arbelos

t_n = 2r / (n2 + 2)

and, for the big one,

r_n = 3r·1/2 / ((n/2)2 + 1/2 + 1) = 6r / (n2 + 6).

In order to derive at anything resembling 2r/13, we have to somehow get rid of the dependency on n. The easiest way to do that is to pass to the reciprocals:

1 / t_n = (n2 + 2) / 2r and
1 / r_n = (n2 + 6) / 6r

Obviously,

3 / r_n - 1 / t_n = 4 / 2r = 2 / r,

independent of n. The latter can be rewritten as

1 / r_n · (3 - r_n / t_n) = 2 / r,

or

r_n / (3 - r_n / t_n) = r / 2.

Perhaps, this is what the tablet was intended for.

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
   • 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
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

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Copyright ? 1996-2014 Alexander Bogomolny