Three Squares and Two Ellipses

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.

Two Ellipses And Three Squares, a 1837 sangaku

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.

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 (pp. 62, 154-155)

    Write to:

    Charles Babbage Research Center
    P.O. Box 272, St. Norbert Postal Station
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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. A Sushi Morsel
  16. An Old Japanese Theorem
  17. Archimedes Twins in the Edo Period
  18. Arithmetic Mean Sangaku
  19. Bottema Shatters Japan's Seclusion
  20. Chain of Circles on a Chord
  21. Circles and Semicircles in Rectangle
  22. Circles in a Circular Segment
  23. Circles Lined on the Legs of a Right Triangle
  24. Equal Incircles Theorem
  25. Equilateral Triangle, Straight Line and Tangent Circles
  26. Equilateral Triangles and Incircles in a Square
  27. Five Incircles in a Square
  28. Four Hinged Squares
  29. Four Incircles in Equilateral Triangle
  30. Gion Shrine Problem
  31. Harmonic Mean Sangaku
  32. Heron's Problem
  33. In the Wasan Spirit
  34. Incenters in Cyclic Quadrilateral
  35. Japanese Art and Mathematics
  36. Malfatti's Problem
  37. Maximal Properties of the Pythagorean Relation
  38. Neuberg Sangaku
  39. Out of Pentagon Sangaku
  40. Peacock Tail Sangaku
  41. Pentagon Proportions Sangaku
  42. Proportions in Square
  43. Pythagoras and Vecten Break Japan's Isolation
  44. Radius of a Circle by Paper Folding
  45. Review of Sacred Mathematics
  46. Sangaku à la V. Thebault
  47. Sangaku and The Egyptian Triangle
  48. Sangaku in a Square
  49. Sangaku Iterations, Is it Wasan?
  50. Sangaku with 8 Circles
  51. Sangaku with Angle between a Tangent and a Chord
  52. Sangaku with Quadratic Optimization
  53. Sangaku with Three Mixtilinear Circles
  54. Sangaku with Versines
  55. Sangakus with a Mixtilinear Circle
  56. Sequences of Touching Circles
  57. Square and Circle in a Gothic Cupola
  58. Steiner's Sangaku
  59. Tangent Circles and an Isosceles Triangle
  60. The Squinting Eyes Theorem
  61. Three Incircles In a Right Triangle
  62. Three Squares and Two Ellipses
  63. Three Tangent Circles Sangaku
  64. Triangles, Squares and Areas from Temple Geometry
  65. Two Arbelos, Two Chains
  66. Two Circles in an Angle
  67. Two Sangaku with Equal Incircles
  68. Another Sangaku in Square
  69. Sangaku via Peru
  70. FJG Capitan's Sangaku

Conic Sections > Ellipse

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