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Harmonic Mean Sangaku: What Is It About?
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Copyright © 1996-2009 Alexander Bogomolny
Harmonic Mean Sangaku
The applet purports to suggest that in a configuration of two vertical segments AE, BD (or two ladders AD, BE inclined at the opposite walls AE, BD), the intersection P of the diagonals is at the height that depends solely on AE and BD but not the distance between the walls. In fact more is true
| 1/CP = 1/AE + 1/BD. |
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Let's denote AC = x and BC = y. Then from similar triangles ABD and ACP we have the proportion
| (x + y)/x = BD/CP = 1 + y/x, |
so that
| y/x = BD/CP - 1 = (BD - CP)/CP. |
In the same way from similar triangles ABE and CBP we obtain
| x/y = AE/CP - 1 = (AE - CP)/CP. |
Multiplying the two yields
| (AE - CP)(BD - CP) = CP2. |
Multiplying through we get after simplification
| AE·BD = CP·(AE + BD). |
A division by AE·BD·CP gives the desired result:
| 1/CP = 1/AE + 1/BD, |
which says that CP is half of the harmonic mean of AE and BD.
References
- H. Fukagawa, D. Pedoe, Japanese Temple Geometry Problems, The Charles Babbage Research Center, Winnipeg, 1989, p. 48
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 49th 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
- 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
- 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 Three Mixtilinear Circles
- Sangaku with Versines
- Sangakus with a Mixtilinear Circle
- Sequences of Touching Circles
- Square and Circle in a Gothic Cupola
- Tangent Circles and an Isosceles Triangle
- The Squinting Eyes Theorem
- Steiner's Sangaku
- 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
Copyright © 1996-2009 Alexander Bogomolny
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