A Sangaku with Many Circles and Some: What Is This About?
A Mathematical Droodle
What if applet does not run? |
|Activities| |Contact| |Front page| |Contents| |Geometry|
Copyright © 1996-2018 Alexander Bogomolny
The applet purports to suggest the following 1837 sangaku [Temple Geometry, 1.5.7] from the Aichi prefecture:
AB is a diameter of O(r) and forms two semicircles. In one of these the equal circles O_{1}(r_{1}) and O'_{1}(r_{1}) touch AB and also O(r) internally. The circles O_{2}(r_{1}) and O'_{2}(r_{1}) are mirror images of these two circles in AB. The circle O(r_{3}) touches the 4 circles we have just described externally, and the circle O_{4}(r_{4}) touches O_{1}(r_{1}) and O'_{1}(r_{1}) externally and O(r) internally. Show that
In the particular case where r_{3} = r/9, show that
where O_{5}(r_{5}) has interior contact, as shown, with both O_{1}(r_{1}) and O'_{1}(r_{1}). |
We'll proceed a step at a time. First, let X be the points of contact of O_{1}(r_{1}) with AB and x = OX.
By the Pythagorean theorem,
(r - r_{1})² = x² + (r_{1})². |
From which
(1) | r_{1} = (r² - x²) / 2r. |
Next we find r_{3}:
(r_{3} + r_{1})² = x² + (r_{1})². |
implying a quadratic equation
(r_{3})² + 2r_{1}r_{3} - x² = 0. |
The equation has two real roots, a negative one and a positive one, the latter being
r_{3} = - r_{1} + √(r_{1}² + x²) = 0. |
Substituting r_{1} from (1) gives
(2) | r_{3} = x² / r. |
Next in line is r_{4}:
for which the Pythagorean theorem provides
(r - r_{1} - r_{4})² + x² = (r_{1} + r_{4})². |
One simplification is immediate:
r² + x² = 2r(r_{1} + r_{4}), |
Substituting r_{1} from (1) gives
r² + x² = r² - x² + 2r·r_{4}, |
from which
(2) | r_{4} = x² / r. |
Thus we see that indeed r_{3} = r_{4}.
When x = r/3, r_{3} = r_{4} = r/9. Note that, for this value of x, r_{1} = 4r/9. Also, circles O_{1}(r_{1}) and O'_{1}(r_{1}) have centers 2x = 2r/3 apart giving the width of the lens shaped intersection as
2r_{1} - 2x = 8r/9 - 2r/3 = 2r/9, |
implying r_{5} = r/9 = r_{3} = r_{4}, for x = r/3.
The sangaku ends here, but having a dynamic applet invites an investigation. If the three equal circles grow while the quadruplets become small, the former eventually form 5 regions, into which one may want to inscribed circles:
When the five circles are equal, their common radius is r/5; obviously. More can be said, viz., the outer four circles are always equal. Furthermore, their radius is exactly r_{1} increasing the number of equal circles to 8. The proof is not difficult and is left as an exercise.
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- H. Fukagawa, A. Rothman, Sacred Geometry: Japanese Temple Geometry, Princeton University Press, 2008, p. 101
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
|Activities| |Contact| |Front page| |Contents| |Geometry|
Copyright © 1996-2018 Alexander Bogomolny
65652150