# 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 S_{1}(3r) and S_{2}(2r) are drawn on AT and BT respectively as diameters. We consider the chain of contact circles O_{i}(r_{i}), _{1}(r_{1}) touches C_{1}(r), drawn on AB as diameter, touches S_{1}(3r) internally and S_{2}(2r) externally, and so on. We also use the circles C_{2}(r) and C_{3}(r) with respective diameters BC and CT to construct another chain of contact circles T_{i}(t_{i}),

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 _{n} < r_{n},*in the spirit* of the required one.

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Copyright © 1996-2017 Alexander Bogomolny

Points T, A, B, C are collinear and AB = BC = CT = 2r. Circles S_{1}(3r) and S_{2}(2r) are drawn on AT and BT respectively as diameters. We consider the chain of contact circles O_{i}(r_{i}), _{1}(r_{1}) touches C_{1}(r), drawn on AB as diameter, touches S_{1}(3r) internally and S_{2}(2r) externally, and so on. We also use the circles C_{2}(r) and C_{3}(r) with respective diameters BC and CT to construct another chain of contact circles T_{i}(t_{i}),

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

- Sangaku: Reflections on the Phenomenon
- Critique of My View and a Response
- 1 + 27 = 12 + 16 Sangaku
- 3-4-5 Triangle by a Kid
- 7 = 2 + 5 Sangaku
- A 49
^{th}Degree Challenge - A Geometric Mean Sangaku
- A Hard but Important Sangaku
- A Restored Sangaku Problem
- A Sangaku: Two Unrelated Circles
- A Sangaku by a Teen
- A Sangaku Follow-Up on an Archimedes' Lemma
- A Sangaku with an Egyptian Attachment
- A Sangaku with Many Circles and Some
- A Sushi Morsel
- An Old Japanese Theorem
- Archimedes Twins in the Edo Period
- Arithmetic Mean Sangaku
- Bottema Shatters Japan's Seclusion
- Chain of Circles on a Chord
- Circles and Semicircles in Rectangle
- Circles in a Circular Segment
- Circles Lined on the Legs of a Right Triangle
- Equal Incircles Theorem
- Equilateral Triangle, Straight Line and Tangent Circles
- Equilateral Triangles and Incircles in a Square
- Five Incircles in a Square
- Four Hinged Squares
- Four Incircles in Equilateral Triangle
- Gion Shrine Problem
- Harmonic Mean Sangaku
- Heron's Problem
- In the Wasan Spirit
- Incenters in Cyclic Quadrilateral
- Japanese Art and Mathematics
- Malfatti's Problem
- Maximal Properties of the Pythagorean Relation
- Neuberg Sangaku
- Out of Pentagon Sangaku
- Peacock Tail Sangaku
- Pentagon Proportions Sangaku
- Proportions in Square
- Pythagoras and Vecten Break Japan's Isolation
- Radius of a Circle by Paper Folding
- Review of Sacred Mathematics
- Sangaku à la V. Thebault
- Sangaku and The Egyptian Triangle
- Sangaku in a Square
- Sangaku Iterations, Is it Wasan?
- Sangaku with 8 Circles
- Sangaku with Angle between a Tangent and a Chord
- Sangaku with Quadratic Optimization
- Sangaku with Three Mixtilinear Circles
- Sangaku with Versines
- Sangakus with a Mixtilinear Circle
- Sequences of Touching Circles
- Square and Circle in a Gothic Cupola
- Steiner's Sangaku
- Tangent Circles and an Isosceles Triangle
- The Squinting Eyes Theorem
- Three Incircles In a Right Triangle
- Three Squares and Two Ellipses
- Three Tangent Circles Sangaku
- Triangles, Squares and Areas from Temple Geometry
- Two Arbelos, Two Chains
- Two Circles in an Angle
- Two Sangaku with Equal Incircles
- Another Sangaku in Square
- Sangaku via Peru
- FJG Capitan's Sangaku

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Copyright © 1996-2017 Alexander Bogomolny

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