Circles and Semicircles in Rectangle: What Is This About?
A Mathematical Droodle

As in one other case, I can't point to a definite source of the problem below. I stumbled upon it at the exceptional site Archimedes' Laboratory where the source was not mentioned. However, the problem is clearly in the Wasan spirit and resembles other sangaku problems.

Created with GeoGebra

The task is apparently to determine the ratio of the sides of the rectangle when the three small circles have the same radius.

Solution

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Copyright © 1996-2017 Alexander Bogomolny

When the three small circles are congruent we arrive at the following diagram:

Assuming the long side of length 2a, the short side 2b and the radius of the small circles r, the Pythagorean theorem applied to one of the four right triangles with the right angle at the center of the diagram leads to the following equation:

(1) b² + (a - r)² = (a + r)².

From which

r = b²/4a.

Since (observing the vertical midline) also r = a - b we obtain a quadratic equation linking a and b:

b² + 4ab - 4a² = 0,

solving which gives

b / a = 2 (2 - 1).

or

a / b = (2 + 1) / 2.

Now note that since r = a - b, b = a - r which says that in the diagram the triangles are isosceles and what looks like a square is a square indeed.

Sangaku

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Copyright © 1996-2017 Alexander Bogomolny

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