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

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.

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 r_i; moreover, denote the sum of distances in the Carnot’s theorem as OO_i. Then, observing that every triangle in the triangulation has the given circle as its circumcircle, Carnot’s thereom states

and the sum in question is given by

Note now that the sum OO₁ + OO₂ + … + OO_n-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.

(There is another theorem that is also known as Japanese.)

References

1. R. Honsberger, Mathematical Gems III, MAA, 1985

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

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