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 Oi(ri), 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 Ti(ti), i = 1, 2, ..., as the figure makes clear. Prove that

tn / (tn / rn - 3) = 2r / 13.

Note that the required identity is likely to be faulty, for it's quite obvious from the construction that tn <  rn, 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 © 1996-2017 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 Oi(ri), 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 Ti(ti), i = 1, 2, ..., as the figure makes clear. Prove that

tn / (tn / rn - 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

tn = 2r / (n² + 2)

and, for the big one,

rn = 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 / tn = (n² + 2) / 2r and
1 / rn = (n² + 6) / 6r

Obviously,

3 / rn - 1 / tn = 4 / 2r = 2 / r,

independent of n. The latter can be rewritten as

1 / rn · (3 - rn / tn) = 2 / r,

or

rn / (3 - rn / tn) = 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 49th Degree Challenge
  7. A Geometric Mean Sangaku
  8. A Hard but Important Sangaku
  9. A Restored 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
  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
  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
  68. Another Sangaku in Square
  69. Sangaku via Peru
  70. FJG Capitan's Sangaku

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Copyright © 1996-2017 Alexander Bogomolny

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