Maximal Properties of the Pythagorean Relation

A 1893 sangaku from the Fukusima prefecture [Fukagawa, Pedoe, problem 2.3.6] admits a relatively straightforward computational solution, but also an elegant solution that practically avoids any kinds of computations.

Rectangles cut off from a right triangle leave three triangles with inradii r1, r2, r3, in the increasing order. Show that when the area of the rectangle is a maximum possible then

(r1)² + (r2)² = (r3)².



  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
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Copyright © 1996-2018 Alexander Bogomolny

Rectangles cut off from a right triangle leave three triangles with inradii r1, r2, r3, in the increasing order. Show that when the area of the rectangle is a maximum possible then

(r1)² + (r2)² = (r3)².


The four triangles ABC, EDC, AEF, DBG are obviously similar. Their inradii relate exactly like any of the corresponding sides. For the definiteness sake, let's focus on the hypotenuse.

If the sides of ΔABC are a = BC, b = AC, c = AB, then, for some k > 0, the corresponding sides in smaller triangles are:

  ABC a b c
  EDC ka kb kc
  AEF ... ... (1-k)b
  DBG ... ... (1-k)a

We want to show that, for k, for which the area of the rectangle DEFG is a maximum,

[(1 - k)a]² + [(1 - k)b]² = (kc)²,


(1) (1 - k)²·a² + (1 - k)²·b² = k²·c².

However, we also have

a² + b² = c².

Multiplying (2) by k² and subtracting from (1) we get

(1 - 2k)·(a² + b²) = 0

which is only possible when k = 1/2. This is then bound to be the optimal configuration implied by the sangaku. To proceed, we need to fill some of the empty cells in the table above:

DG / AC = BD / AB,

so that DG = (1 - k)ab/c. Since DE = kc, the area S of the rectangle DEFG is given by

S = DG × DE = (1 - k)k·ab

which attains its maximum along with f(k) = (1 - k)k. This is an inverted parabola with the roots at 0 and 1, and the maximum the midway between the two, i.e., at k = 1/2.


  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

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Copyright © 1996-2018 Alexander Bogomolny