Arithmetic Mean Sangaku: What Is This About?
A Mathematical Droodle

Solution

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

The applet is hopefully suggestive of the following sangaku [Bicycle, #39]:

Points C and D lie on a circle with diameter AB, CD AB. One circle is inscribed into ΔABC. Two more circles inscribed on the other side of AB in the curvilinear triangles formed by AB, CD and the given circle. Prove that the radius of the former is the average of the radii of the latter. Moreover, the lines through the centers of the latter circles are tangent to the former circle.

We apply the Pythagorean theorem twice to the right triangles with hypotenuses joining the centers of the circles, as shown:

Let the distance from center O of the circumcircle to CD be d, the other centers I, P, Q, and the radii of the circles be R, r, r_L, r_R. Then the Pythagorean theorem gives

From which we express r_R and r_L:

In our case, c = 2R, while a and b can again be found with the help of the Pythagorean theorem. It's not hard to see that one of them is √2R(R + d) whereas the other is √2R(R - d), which conforms with (2) and shows that the horizontal distance between P and Q is 2r. We may rewrite (1) as

(1')	rR = -(R + d) + b rL = -(R - d) + a.

Thus ST = 2r, where S and T are the points of tangency of (P) and (Q) with AB. In particular,

(This sangaku has a nice generalization.)

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. J. Konhauser, D. Velleman, S. Wagon, Which Way Did the Bicycle Go?, MAA, 1996

Copyright © 1996-2008 Alexander Bogomolny