One of the simplest sangaku - geometric problems carved on colorful wooden tablets and offered in shinto shrines and buddhist temples during the Edo period (1603-1867) of self-imposed seclusion of Japan from the Western world - is presented by the following diagram:

A triangle is formed by a line that joins the base of a square with the midpoint of the opposite side and a diagonal. Find the radius of the inscribed circle.

Solution

Copyright © 1996-2009 Alexander Bogomolny

A triangle is formed by two lines that join the base of a square with the midpoint of the opposite side and a diagonal. Find the radius of the inscribed circle.

A solution could be observed on an extended diagram. Let A, B, C, D be the vertices of the square and M the midpoint of AD. BD and CM intersect in P. Let Q be the intersection of BM and CD and H the projection of P on BC. The task is to find the inradius of ΔBCP.

Focus on ΔBCQ. MD is parallel to the base and is half as long which implies that this is a midline of the triangle. In other words, M and D are the midpoints of BQ and CQ, respectively. This means that BD and CM are two medians in the triangle and P its centroid. The centroid divides the medians in ratio 2:1 so that

(1)	CP = CM·2/3 and BP = BD·2/3.

Thus assuming BC = a, we can apply the Pythagorean theorem to first find BD (from ΔBCD) and CM (from ΔCDM) and subsequently, from (1) the sides of ΔBCP. By the same token, altitude PH = a·2/3. But in any triangle,

which is quite in agreement with the exact answer (3). The difference between the two is that the numeric value, however accurate, carries no information as to the manner in which it was obtained. In (3), the appearance of the square roots is suggestive and points to a possible application of the Pythagorean theorem to two right triangles, as we did above. Thus (3) exhibits a valuable pattern that may trigger a thought process that may lead to the recollection of properties of medians and centroids, whereas r = .2480006466 in itself is a naked approximate number that, for all we know, is at best close enough to the exact answer.

In short, this 300 year old problem serves an example where the use of calculators actually obscures relationships between numbers. More such examples can be found in an edifying book What The Numbers Say by D. Niederman and D. Boyum along with a multitude of observations on everyday number usage and the ability to see beyond the numbers.

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

2. D. Niederman and D. Boyum, What The Numbers Say, Broadway Books, 2003

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 Restored Sangaku Problem
   • A better solution to a difficult 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. Circles and Semicircles in Rectangle
21. Circles in a Circular Segment
22. Circles Lined on the Legs of a Right Triangle
23. Equal Incircles Theorem
24. Equilateral Triangle, Straight Line and Tangent Circles
25. Equilateral Triangles and Incircles in a Square
26. Five Incircles in a Square
27. Four Hinged Squares
28. Four Incircles in Equilateral Triangle
29. Gion Shrine Problem
30. Harmonic Mean Sangaku
31. Heron's Problem
32. In the Wasan Spirit
33. Incenters in Cyclic Quadrilateral
34. Japanese Art and Mathematics
35. Malfatti's Problem
36. Maximal Properties of the Pythagorean Relation
37. Neuberg Sangaku
38. Out of Pentagon Sangaku
39. Peacock Tail Sangaku
40. Pentagon Proportions Sangaku
41. Pythagoras and Vecten Break Japan's Isolation
42. Radius of a Circle by Paper Folding
43. Review of Sacred Mathematics
44. Sangaku à la V. Thebault
45. Sangaku and The Egyptian Triangle
46. Sangaku in a Square
47. Sangaku Iterations, Is it Wasan?
48. Sangaku with 8 Circles
49. Sangaku with Three Mixtilinear Circles
50. Sangaku with Versines
51. Sangakus with a Mixtilinear Circle
52. Sequences of Touching Circles
53. Square and Circle in a Gothic Cupola
54. Tangent Circles and an Isosceles Triangle
55. The Squinting Eyes Theorem
56. Steiner's Sangaku
57. Three Incircles In a Right Triangle
58. Three Squares and Two Ellipses
59. Three Tangent Circles Sangaku
60. Triangles, Squares and Areas from Temple Geometry
61. Two Arbelos, Two Chains
62. Two Circles in an Angle
63. Two Sangaku with Equal Incircles

Copyright © 1996-2009 Alexander Bogomolny