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 © 1996-2008 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 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 ŕ 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

Copyright © 1996-2008 Alexander Bogomolny