A Geometric Mean Sangaku
It is often intimated that some of the sangaku - geometric problems carved on colorful wooden tablets - have been posted by clever teenagers. Of course, 200-400 years later, no one can be certain about that, but if there was indeed a problem suitable for an early age, the one below fits the bill (Kiyomizu Temple, Kyoto Prefecture).
Squares and circles are inscribed in successive right triangles. What is the relation between the radii of the circles?
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Copyright © 1996-2018 Alexander Bogomolny
When a square is inscribed in a right triangle with one side on the hypotenuse of the latter, it cuts off three smaller triangles all similar to the initial one. In the problem, the process is repeated with the largest of the three and then again with the largest of the newly created pieces. In all cases, a square is inscribed into similar triangles.
So again, let there be a right triangle with hypotenuse c_{1} and c_{2} be the hypotenuse of the largest of the three pieces cut off from the triangle by the inscribed square.
c_{2} appears to be the mean proportional between c_{1} and c_{3}. But then again, because of the similarity, the same is true for any linear element in the triangles, not only the hypotenuse. In particular, the radii of the three circles constructed similarly in the similar triangles satisfy the mean proportional condition:
r_{1} : r_{2} = r_{2} : r_{3}. |
In other words, the radius of middle (blue) circle is the geometric mean of the radii of the two red circles.
References
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
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 © 1996-2018 Alexander Bogomolny
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