R. Honsberger,
Mathematical Gems, III,
MAA, 1985, pp. 24-26

An Old Japanese Theorem

In Roger Johnson's marvellous old geometry text-Advanced Euclidean Geometry, first published in 1929 - he reports (on page 193 of the Dover edition, 1960) on the ancient custom by Japanese mathematicians of inscribing their discoveries on tablets which were hung in the temples to the glory of the gods and the honor of the authors. The following gem is known to have been exhibited in this way in the year 1800.

Let a convex polygon, which is inscribed in a circle, be triangulated by drawing all the diagonals from one of the vertices, and let the inscribed circle be drawn in each of the triangles. Then the sum of the radii of all these circles is a constant which is independent of which vertex is used to form the triangulation (Figure 1).

A great deal more might have been claimed, for this same sum results for every way of triangulating the polygon! (Figure 2). As we shall see, a simple application of a beautiful theorem of L. N. M. Carnot (1753-1823) settles the whole affair.

First of all, observe that any triangulation of an n-gon with its diagonals consists of (n - 2) triangles. Next assume that the triangles in the triangulation are numbered, and that the inradius of triangle i is ri; moreover, denote the sum of distances in the Carnot's theorem as OOi. Then, observing that every triangle in the triangulation has the given circle as its circumcircle, Carnot's thereom states

ri + R = OOi

and the sum in question is given by

(*) r1 + r2 + ... + rn-2 = OO1 + OO2 + ... + OOn-2 - (n - 2)R

Note now that the sum OO1 + OO2 + ... + OOn-2 on the right consists of perpendiculars to the sides of the polygon each counted once, and perpendiculars to the diagonals each counted twice since every diagonal serves as a side of two adjacent triangles. Furthermore, considering such internal perpendiculars, each is taken once with the sign plus and another time with the sign minus thus contributing nothing to the sum. Therefore, this sum is actually equal to the sum of all perpendiculars from O to the sides of the polygon. And, therefore, the right-hand side in (*) is constant. So is the left-hand side.

Remark 1

There is another theorem that is also known as Japanese.

Remark 2

David Richeson of Dickinson College has extended (2011) the theorem to non-convex polygons.

References

  1. R. Honsberger, Mathematical Gems III, MAA, 1985
  2. R. A. Johnson, Advanced Euclidean Geometry (Modern Geometry), Dover, 1960

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

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