Sangaku with Three Mixtilinear Circles

Sangaku traditionally contained a diagram and a question concerning the diagram. Sometimes there were also instructions for constructing of the depicted objects and occasionally a solution to the problem. One such sangaku with a solution could be found in [Smith and Mikami, p. 185]:

There is a circle in which a triangle and three circles, A, B, C, are inscribed in the manner shown in the figure. Given the diameters of the three inscribed circles, required the diameter of the circumscribed circle.

(The three inscribed circles are the mixtilinear circles in the inscribed triangle. Each mixtilinear circle is inscribed in an angle and touches the circumcircle of the triangle. In the applet below, I labeled A, B, C the vertices of the triangle with understanding that the circles are also labeled A, B, C depending on the angle into which each is inscribed.)

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at, download and install Java VM and enjoy the applet.

What if applet does not run?

The sangaku comes with a solution that is representative of the contemporary mathematical style:

Let the respective diameters be x, y, and z, and let xy = a. Then from a² take [(x - y)z]². Divide a by the remainder and call the result b. Then from (x + y)z take a and divide 0.5 by this remainder and add b, and then multiply by z and by a. The result is the diameter of the circumscribed circle.

Smith and Mikami note that to this rule is appended, with some note of pride, the words: "Feudal District of Kakegawa in Yenshu Province, third month of 1795, Miyajima Sonobei Keichi, pupil of Fujita Sadasuke of the School of Seki."

The rule can be translated as

xyz [xy / (x²y² - (x - yx²) + 0.5 /((x + y)z - xy)]

The applet let's you verify that the rule does work.


  1. D. E. Smith and Yoshio Mikami, A History of Japanese Mathematics, Dover, 2004 (originally 1914)
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