Five Incircles in a Square: What Is This About?
A Mathematical Droodle

Explanation

Copyright © 1996-2008 Alexander Bogomolny

Here we have a very characteristic sangaku - a Temple geometry - problem. Most of geometric sangaku dealt with several circles inscribed in other circles or other shapes. In addition, most often a sangaku was a computational problem as opposed to problems that require a proof. The problem below is a little of both: a proof of the result is supported by a chain of calculations.

In the manner of proof #3 of the Pythagorean theorem, four equal right triangles and a small square are combined into a larger square. Circles are inscribed into the four triangles and the inner square. A question: What can be said of the configuration where all five circles have equal radii?

So assume the triangles have legs a and b (a > b) and hypotenuse c. Then, as we know, the inradius for the triangle is given by

On the other hand, the radius of the circle inscribed into the inner square is obviously

Equating the two we get

which means that the angle opposite b is 30° and the other one is 60°. From here we easily find the common inradius: r = c·(√3 - 1)/4.

Most of the sangaku problems are much more difficult.

References

1. H. Fukagawa, D. Pedoe, Japanese Temple Geometry Problems, The Charles Babbage Research Center, Winnipeg, 1989

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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