This is another in a series of simple sangaku problems that is solved by a repeated application of the Pythagorean theorem.

Two quarter circles inscribed in a square form a gothic cupola. Inscribed in the latter is a circle on top which stands a small circle. What is the relation between the radius of the circle and the side of the small square and the side the big square?

Solution

Copyright © 1996-2008 Alexander Bogomolny

With the reference to the diagram, assume the side of the big square is 1, x the side of the small square and r is the radius of the circle in question.

In right ΔAFG, AG² = AF² + FG²:

1 = (1/2 + x/2)2 + x2,

which simplifies to a quadratic equation

5x2 + 2x - 3 = 0,

with a single positive root, x = 3/5. The Pythagorean theorem applied to ΔAMO, AO² = AM² + MO², supplies an equation for the radius r:

(1 - r)2 = (1/2)2 + (x + r)2,

which, with x = 3/5, simplifies to a linear equation in r so that r = 39/320.

References

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

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6. A 49^th Degree Challenge
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13. A Sangaku with Many Circles and Some
14. An Old Japanese Theorem
15. Archimedes Twins in the Edo Period
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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
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32. Japanese Art and Mathematics
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35. Neuberg Sangaku
36. Out of Pentagon Sangaku
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39. Pythagoras and Vecten Break Japan's Isolation
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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
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53. The Squinting Eyes Theorem
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56. Three Squares and Two Ellipses
57. Three Tangent Circles Sangaku
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60. Two Circles in an Angle

Copyright © 1996-2008 Alexander Bogomolny