Equilateral Triangle, Straight Line and Tangent Circles: What Is This About?
A Mathematical Droodle

Solution

Copyright © 1996-2009 Alexander Bogomolny

The applet purports to suggest the following sangaku [Temple Geometry, p. 25, #2.1.11]:

(This is a 1885 Sangaku from the Fukusima prefecture whose tablet has disappeared long ago.)

Let K, L, M, N, S, T be the points of tangency as shown in the applet and s be the common length of the sides of ΔABC. Denote

Then

Two external tangents to a pair of circles are equal: KL = ST which tells us that

implying

But triangles OAK and QBL are both 30°-60°-90°. Therefore, r_a = √3·x and r_b = √3·y such that

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
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6. A 49^th Degree Challenge
7. A Geometric Mean Sangaku
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13. A Sangaku with an Egyptian Attachment
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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
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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?
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49. Sangaku with Three Mixtilinear Circles
50. Sangaku with Versines
51. Sangakus with a Mixtilinear Circle
52. Sequences of Touching Circles
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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
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62. Two Circles in an Angle
63. Two Sangaku with Equal Incircles

Copyright © 1996-2009 Alexander Bogomolny