Circles Lined on the Legs of a Right Triangle: What Is This About?
A Mathematical Droodle
What if applet does not run? |
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Copyright © 1996-2018 Alexander Bogomolny
The applet purports to suggest an extension of the following sangaku [Temple Geometry, p. 36]:
ABC is a right-angled triangle at C, and 8 circles of the same radius r lie within the triangle and have various contacts. O_{1}(r) touches BC and AB. O_{2}(r) touches BC and O_{1}(r), O_{3}(r) touches BC and O_{2}(r), O_{4}(r) touches BC and O_{3}(r), and O_{5}(r) touches both BC and AC and also O_{4}(r). O_{6}(r) touches AC and O_{5}(r), O_{7}(r) touches AC and O_{6}(r), and finally O_{8}(r) touches AC and AB and O_{7}(r). Show that the radius of the incircle of triangle ABC is equal to 3r. |
(The problem appears on a surviving 1892 tablet in the Hyogo prefecture. )
Draw a line parallel to the b = AC side of the triangle through the upper most point of tangency of the circles. We get two similar right triangles with legs a and (a - 6r) and inradius, say, R and r. Which gives a proportion
(1) | r/R = (a - 6r) / a = 1 - 6r/a. |
For a triangle similarly obtained in the A corner we have
r/R = (b - 8r) / b = 1 - 8r/b. |
Comparing the two expressions we see that
8a = 6b or 4a = 3b. |
If, for example, a = 3 then b = 4 and the hypotenuse equals 5 giving a 3-4-5 triangle. The inradius can be found from
R = (a + b - c)/2, |
or, in this case, R = (3 + 4 - 5)/2 = 1. On the other hand, from say (1),
r = aR / (a + 6R), |
so that r = 1/3 and, indeed, R = 3r, as required.
As the applet shows (and calculations are entirely generic), any Pythagorean triple
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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