# Sangaku with Three Mixtilinear Circles

Sangaku traditionally contained a diagram and a question concerning the diagram. Sometimes there were also instructions for constructing of the depicted objects and occasionally a solution to the problem. One such sangaku with a solution could be found in [Smith and Mikami, p. 185]:

There is a circle in which a triangle and three circles, A, B, C, are inscribed in the manner shown in the figure. Given the diameters of the three inscribed circles, required the diameter of the circumscribed circle.

(The three inscribed circles are the mixtilinear circles in the inscribed triangle. Each mixtilinear circle is inscribed in an angle and touches the circumcircle of the triangle. In the applet below, I labeled A, B, C the vertices of the triangle with understanding that the circles are also labeled A, B, C depending on the angle into which each is inscribed.)

What if applet does not run? |

The sangaku comes with a solution that is representative of the contemporary mathematical style:

Let the respective diameters be *x*, *y*, and *z*, and let *xy* = *a*.*a*² take *x* - *y*)*z*]².*a* by the remainder and call the result *b*. Then from *x* + *y*)*z**a* and divide 0.5 by this remainder and add *b*, and then multiply by *z* and by *a*. The result is the diameter of the circumscribed circle.

Smith and Mikami note that to this rule is appended, with some note of pride, the words: "Feudal District of Kakegawa in Yenshu Province, third month of 1795, Miyajima Sonobei Keichi, pupil of Fujita Sadasuke of the School of Seki."

The rule can be translated as

*xyz* [*xy* / (*x*²*y*² - (*x* - *y*)²*x*²) + 0.5 /((*x* + *y*)*z* - *xy*)]

The applet let's you verify that the rule does work.

### References

- D. E. Smith and Yoshio Mikami,
*A History of Japanese Mathematics*, Dover, 2004 (originally 1914)

## 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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