Hiyoshi Shrine is located 3.4km north of Kamogata Station on the JR Sanyo Main Line.
Problem
As shown in the figure, four large circles intersect at one point, and four small circles are in between. What is the diameter of the small circle when the diameter of the large circle is 3 inches?
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$\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$
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Solution
Let $d$ be the diameter of the large circle and $x$ be the diameter of the small circle.
When the intersection of the four large circles is $O$, the center of the upper large circle is $O’$ and the center of the right small circle is $O^”$, then $∆O’OO^”$ is a rectangular triangle. Therefore from the Pythagorean theorem, it follows that
$$(O’O)^2+(OO^”)^2=(O^”O’)^2,$$
$$(\frac{d}{2})^2+(d-\frac{x}{2})^2=(\frac{x}{2}+\frac{d}{2})^2,$$
$$d^2+(2d-x)^2=(x+d)^2,$$
$$6dx-4d^2=0,$$
$$(6x-4d)d=0,$$
$$6x-4d=0$$
$(\because d\neq0),$
$$\therefore \ x=\frac{2}{3}d.$$
Since $d=3$ from the problem statement, we see that
$$x=\frac{2}{3}\times3=2.$$
$$(Answer) \ 2 \ inches.$$
Reference
Yoshikazu Yamakawa, ed. (1997) Okayama ken no Sangaku (Sangaku in Okayama Prefecture), pp.59-60; pp.285-286.
27 responses to “Hiyoshi Shrine (1940), Kamogata-cho, Asakuchi City, Okayama Prefecture”
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