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Three Squares and Two Ellipses

The applet bellow serves to illustrate the following sangaku [Temple Geometry, pp. 62, 154-155]:

 

Two congruent ellipses with the major axis 2a and the minor axis 2b touch each other, the minor axis of each passing through the point of contact; l is a common exterior tangent parallel to the minor axis. There are three congruent squares of side t such that two of the squares have a side in common and another side lying on l, and each has a vertex on one of the ellipses, and a third square has a vertex on each ellipse and a side that lies along a side of the first two squares.

Show that

  a = 2b and t = a/5.

 

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


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The problem was carved on an 1838 tablet in the Aichi prefecture. The tablet has since disappeared, but the solution has been discussed in 1976 by H. Fukagawa with a reference to a 1837(!) publication, Chosyu Sinpeki by Tameyuki Yosida. The title translates as Tablets in the Chosyu Prefecture.

Solution

References

  1. 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

Copyright © 1996-2009 Alexander Bogomolny

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Solution

The calculations that are necessarily a part of any solution are greatly simplified by an affine trasnformation that squeezes the diagram towards l with coefficient b/a. The horizontal dimensions remain the same, whereas all vertical dimensions are mulitplied by b/a. In particular, the ellipses convert to the circles of radius b and the t×t squares become rectangles t'×t, with t' = tb/a.

 

In the diagram,

  q = b - t', p = b - 2t', u = b - t, v = b - t/2.

From

  b² = q² + u² = (b - bt/a)² + (b - t)²

it follows that

(1) 2bt²/a = b² - 2bt(1 + b/a) + t²(1 + b/a)²

so that (very clever, I must say)

(1') t 2b/a  = b - (1 + b/a)t.

Again, from

  b² = v² + p² = (b - t/2)² + (b - 2bt/a)²

we can deduce that

(2) 2bt²/a = b² - 2bt(1/2 + 2b/a) + t²(1/2 + 2b/a)²

so that

(2') t 2b/a  = b - (1/2 + 2b/a)t.

From (1') and (2'),

  1 + b/a = 1/2 + 2b/a,

that is, a = 2b. From (1'), t = a/2 - 3t/2, or 5t = a.

Note: the steps from (1) to (1') and from (2) to (2') are based on unproven assumptions that the right hand sides in (1) and (2) are positive. If they are not, the derivations amount to claiming that 5² = (-5)², say, implies 5 = -5. Which is of course absurd.

Sangaku

  1. Sangaku: Reflections on the Phenomenon
  2. Critique of My View and a Response
  3. 1 + 27 = 12 + 16 Sangaku
  4. 3-4-5 Triangle by a Kid
  5. 7 = 2 + 5 Sangaku
  6. A 49th Degree Challenge
  7. A Geometric Mean Sangaku
  8. A Hard but Important Sangaku
  9. A Restored Sangaku Problem
  10. A Sangaku: Two Unrelated Circles
  11. A Sangaku by a Teen
  12. A Sangaku Follow-Up on an Archimedes' Lemma
  13. A Sangaku with an Egyptian Attachment
  14. A Sangaku with Many Circles and Some
  15. An Old Japanese Theorem
  16. Archimedes Twins in the Edo Period
  17. Arithmetic Mean Sangaku
  18. Bottema Shatters Japan's Seclusion
  19. Circles and Semicircles in Rectangle
  20. Circles in a Circular Segment
  21. Circles Lined on the Legs of a Right Triangle
  22. Equal Incircles Theorem
  23. Equilateral Triangle, Straight Line and Tangent Circles
  24. Equilateral Triangles and Incircles in a Square
  25. Five Incircles in a Square
  26. Four Hinged Squares
  27. Four Incircles in Equilateral Triangle
  28. Gion Shrine Problem
  29. Harmonic Mean Sangaku
  30. Heron's Problem
  31. In the Wasan Spirit
  32. Incenters in Cyclic Quadrilateral
  33. Japanese Art and Mathematics
  34. Malfatti's Problem
  35. Maximal Properties of the Pythagorean Relation
  36. Neuberg Sangaku
  37. Out of Pentagon Sangaku
  38. Peacock Tail Sangaku
  39. Pentagon Proportions Sangaku
  40. Pythagoras and Vecten Break Japan's Isolation
  41. Radius of a Circle by Paper Folding
  42. Review of Sacred Mathematics
  43. Sangaku à la V. Thebault
  44. Sangaku and The Egyptian Triangle
  45. Sangaku in a Square
  46. Sangaku Iterations, Is it Wasan?
  47. Sangaku with 8 Circles
  48. Sangaku with Three Mixtilinear Circles
  49. Sangaku with Versines
  50. Sangakus with a Mixtilinear Circle
  51. Sequences of Touching Circles
  52. Square and Circle in a Gothic Cupola
  53. Tangent Circles and an Isosceles Triangle
  54. The Squinting Eyes Theorem
  55. Steiner's Sangaku
  56. Three Incircles In a Right Triangle
  57. Three Squares and Two Ellipses
  58. Three Tangent Circles Sangaku
  59. Triangles, Squares and Areas from Temple Geometry
  60. Two Arbelos, Two Chains
  61. Two Circles in an Angle
  62. Two Sangaku with Equal Incircles

Conic Sections > Ellipse

Copyright © 1996-2009 Alexander Bogomolny

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