It is often intimated that some of the sangaku - geometric problems carved on colorful wooden tablets - have been posted by clever teenagers. Of course, 200-400 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?

Solution

Copyright © 1996-2009 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 c₁ and c₂ be the hypotenuse of the largest of the three pieces cut off from the triangle by the inscribed square. q = c₁ / c₂. If we repeat the operation with the largest triangle we'll get a smaller triangle of hypotenuse c₃ such that q = c₂ / c₃, because of the similarity.

c₂ appears to be the mean proportional between c₁ and c₃. 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:

In other words, the radius of middle (blue) circle is the geometric mean of the radii of the two red circles.

References

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

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Copyright © 1996-2009 Alexander Bogomolny