Problem
As shown in the figure, two large circles and two small circles are inscribed in a square, separated by oblique lines. If the diameter of the large circle is $1 \ inch$, what is the diameter of the small circle?
$$ $$
$$ $$
$\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$
$$ $$
$$ $$
$$ $$
$$ $$
$$ $$
$$ $$
$$ $$
$$ $$
$$ $$
$$ $$
Solution
Let the diameter of the large circle be $l$, and the diameter of the small circle be $s$.
From the figure, the diagonals of this square can be expressed in two ways:
$$AC=\frac{\sqrt{2} s}{2}+\frac{s}{2}+l+\frac{s}{2}+\frac{\sqrt{2} s}{2}=l+(\sqrt{2}+1)s,$$
$$BD=\frac{\sqrt{2} l}{2}+l+\frac{\sqrt{2} l}{2}=(\sqrt{2}+1)l.$$
Since $AC=BD$,
$$l+(\sqrt{2}+1)s=(\sqrt{2}+1)l,$$
$$∴ \ s=\frac{\sqrt{2} l}{\sqrt{2}+1}=(2-\sqrt{2})l. \qquad [*]$$
By substituting $l=1$ for $[*]$, we get
$$s=2-\sqrt{2}≒0.5857…$$
Reference
Yoshikazu Yamakawa, ed. (1997) Okayama ken no Sangaku (Sangaku in Okayama Prefecture), p.40; pp.347-348.