Katayama-hiko Shrine is located $850$ meters southeast from Osafune Station on the JR Ako Line.
Problem
As shown in the figure, two mutually circumscribed circles of diameter $d$ are inscribed in a fan of radius $R$.
$c$ is the common tangent of these two circles, and is also the chord of the fan that inscribes these circles, and $t$ is the diameter of the small circle that touches the chord and the arc of the fan at their respective midpoints. Let $c=3.62438 \ inches$ and $t=0.34 \ inches$. Then, find the diameter $d$ of the equal circle.
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Solution
The red right triangle on the left side of the diagram has the following relationship:
$$(\frac{c}{2})^2+(R-t)^2=R^2,$$
$$c^2-8Rt+4t^2=0,$$
$$∴ \ R=\frac{c^2+4t^2}{8t}. \qquad [1]$$
The blue right triangle on the right side of the diagram has the following relationship:
$$(\frac{d}{2})^2+(R-t-\frac{d}{2})^2=(R-\frac{d}{2})^2,$$
$$d^2+4t^2-8Rt+4dt=0,$$
$$∴ \ R=\frac{d^2+4t^2+4dt}{8t}. \qquad [2]$$
From $[1]$ and $[2]$,
$$\frac{c^2+4t^2}{8t}=\frac{d^2+4t^2+4dt}{8t},$$
$$c^2+4t^2=d^2+4t^2+4dt, \qquad (∵ \ 8t≠0)$$
$$d^2+4dt-c^2=0,$$
$$d^2+1.36d-13.1361303844=0,$$
$$∴ d=\frac{-1.36±\sqrt{1.8496+52.5445215376}}{2}≃\frac{-1.36±7.37523704958695}{2}.$$
Since $d>0$,
$$d≃3.00761852479347.$$
$$Answer \quad approximately \ 3.0076 \ inches.$$
Reference
Yoshikazu Yamakawa, ed. (1997) Okayama ken no Sangaku (Sangaku in Okayama Prefecture), pp.37-38; pp.365-367.