Equal Incircles Theorem: What Is It About?
A Mathematical Droodle

(The number of points on the base line which is originally 4 can be changed by clicking on it a little off its center line.)

Explanation

Copyright © 1996-2009 Alexander Bogomolny

The applet suggests the following theorem [Wells, p. 67] from triangle geometry:

I do not know of an elegant proof of that theorem. In [Wells, p. 67] it appears without a proof. There's no reference either.

The problem has been also discussed by R. Honsberger [Delights, section 17], where he mistakenly claims that the equality of the incircles only extends on the "triangles formed by sets of 2ⁿ consecutive triangles in the fan". The solution, nonetheless, works for a more general statement. Honsberger found this to be #2.2.5 in the Sangaku collection by H. Fukagawa and D. Pedoe. In fact problem #2.2.5 shows only two circles and the question is to find the length of the common tangent from the apex A in case the two circles are equal. In this form the problem appears on a surviving 1897 tablet from the Chiba prefecture. The answer to this problem can be found to be

where a is the base and s is the semiperimeter of the triangle.

However, the theorem is a consequence (and also a generalization) of a theorem published in 1986 by H. Demir, which, as was shown by J. B. Tabov, admits if not a more elegant proof, then quite an elegant generalization:

References

1. H. Fukagawa, D. Pedoe, Japanese Temple Geometry Problems, The Charles Babbage Research Center, Winnipeg, 1989
2. R. Honsberger, Mathematical Delights, MAA, 2004
3. D. Wells, Curious and Interesting Geometry, Penguin Books, 1991

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