A Geometric Mean Sangaku
It is often intimated that some of the sangaku  geometric problems carved on colorful wooden tablets  have been posted by clever teenagers. Of course, 200400 years later, no one can be certain about that, but if there was indeed a problem suitable for an early age, the one below fits the bill (Kiyomizu Temple, Kyoto Prefecture).
Squares and circles are inscribed in successive right triangles. What is the relation between the radii of the circles?
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Copyright © 19962018 Alexander Bogomolny
When a square is inscribed in a right triangle with one side on the hypotenuse of the latter, it cuts off three smaller triangles all similar to the initial one. In the problem, the process is repeated with the largest of the three and then again with the largest of the newly created pieces. In all cases, a square is inscribed into similar triangles.
So again, let there be a right triangle with hypotenuse c_{1} and c_{2} be the hypotenuse of the largest of the three pieces cut off from the triangle by the inscribed square.
c_{2} appears to be the mean proportional between c_{1} and c_{3}. But then again, because of the similarity, the same is true for any linear element in the triangles, not only the hypotenuse. In particular, the radii of the three circles constructed similarly in the similar triangles satisfy the mean proportional condition:
r_{1} : r_{2} = r_{2} : r_{3}. 
In other words, the radius of middle (blue) circle is the geometric mean of the radii of the two red circles.
References
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
Winnipeg, MB
Canada R3V 1L6
Sangaku

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Contact Front page Contents Geometry
Copyright © 19962018 Alexander Bogomolny