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

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   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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9. A Restored Sangaku Problem
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11. A Sangaku by a Teen
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13. A Sangaku with an Egyptian Attachment
14. A Sangaku with Many Circles and Some
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16. Archimedes Twins in the Edo Period
17. Arithmetic Mean Sangaku
18. Bottema Shatters Japan's Seclusion
19. Circles and Semicircles in Rectangle
20. Circles in a Circular Segment
21. Circles Lined on the Legs of a Right Triangle
22. Equal Incircles Theorem
23. Equilateral Triangle, Straight Line and Tangent Circles
24. Equilateral Triangles and Incircles in a Square
25. Five Incircles in a Square
26. Four Hinged Squares
27. Four Incircles in Equilateral Triangle
28. Gion Shrine Problem
29. Harmonic Mean Sangaku
30. Heron's Problem
31. In the Wasan Spirit
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33. Japanese Art and Mathematics
34. Malfatti's Problem
35. Maximal Properties of the Pythagorean Relation
36. Neuberg Sangaku
37. Out of Pentagon Sangaku
38. Peacock Tail Sangaku
39. Pentagon Proportions Sangaku
40. Pythagoras and Vecten Break Japan's Isolation
41. Radius of a Circle by Paper Folding
42. Review of Sacred Mathematics
43. Sangaku à la V. Thebault
44. Sangaku and The Egyptian Triangle
45. Sangaku in a Square
46. Sangaku Iterations, Is it Wasan?
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48. Sangaku with Three Mixtilinear Circles
49. Sangaku with Versines
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53. Tangent Circles and an Isosceles Triangle
54. The Squinting Eyes Theorem
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56. Three Incircles In a Right Triangle
57. Three Squares and Two Ellipses
58. Three Tangent Circles Sangaku
59. Triangles, Squares and Areas from Temple Geometry
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62. Two Sangaku with Equal Incircles

Copyright © 1996-2009 Alexander Bogomolny