Problem
If a rectangle contains a large circle, a medium circle, and a small circle, and the diameters of each are $8 ins., 6 ins.$, and $3 ins.$, as shown in the figure, find the length and width of the rectangle.
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Solution
$$LM⊥O’O^{”},$$
$$ON⊥O’O^{”}.$$
$$OO’=\frac{8+6}{2}=7,$$
$$OO^{”}=\frac{8+3}{2}=5.5,$$
$$O’O^{”}=\frac{6+3}{2}=4.5.$$
$$O’O^{”}L〜O’LM,$$
$$O’L=O’j-LJ=\frac{6-3}{2}=1.5,$$
$$O^{”}L=\sqrt{{O’O^{”}}^2-LO’^2}=\sqrt{4.5^2-1.5^2}=\sqrt{18}=3\sqrt{2}.$$
$$\frac{LM}{O’L}=\frac{O^{”}L}{O’O^{”}}, \qquad (∵ O’O^{”}L〜O’LM)$$
$$∴\quad LM=\frac{O^{”}L×O’L}{O’O^{”}}=\frac{3 \sqrt{2}×1.5}{4.5}=\sqrt{2}.$$
$$MO’^2=O’L^2-LM^2=1.5^2-(\sqrt{2})^2=0.25,$$
$$∴ \quad MO’=0.5.$$
Next, regarding the triangle $OO’O^{”}$, from the cosine theorem,
$$OO’^2={OO^{”}}^2+{O’O^{”}}^2-2×OO^{”}×O’O^{”}×cos∠OO^{”}O’,$$
$$7^2=5.5^2+4.5^2-2×5.5×4.5×cos∠OO^{”}O’,$$
$$∴ \quad cos∠OO^{”}O’=\frac{5.5^2+4.5^2-7^2}{2×5.5×4.5}=\frac{1}{33},$$
$$∴ \quad sin∠OO^{”}O’=\sqrt{1-(\frac{1}{33})^2}=\frac{8 \sqrt{17}}{33}.$$
Regarding the right triangle $OO^{”}N$,
$$ON=OO^{”}×sin∠OO^{”}O’=5.5×\frac{8 \sqrt{17}}{33}=\frac{44 \sqrt{17}}{33},$$
$$O^{”}N=OO^{”}×cos∠OO^{”}O’=5.5×\frac{1}{33}=\frac{11}{66}.$$
From $∆EON〜∆O’LM$,
$$\frac{NE}{ON}=\frac{MO’}{LM},$$
$$∴ \quad NE=\frac{MO’×ON}{LM}=\frac{0.5}{\sqrt{2}}×\frac{44 \sqrt{17}}{33}=\frac{11 \sqrt{34}}{33},$$
$$∴ \quad EO^{”}=NE-O^{”}N=\frac{11 \sqrt{34}}{33}-\frac{11}{66}=\frac{22 \sqrt{34}-11}{66}.$$
From $∆EO^{”}F〜∆O’LM$,
$$∠FEO^{”}=∠MO’L,$$
$$sin∠FEO^{”}=\frac{LM}{O’L}=\frac{sqrt{2}}{1.5}=\frac{2 \sqrt{2}}{3},$$
$$∴ \quad O^{”}F=EO^{”}×sin∠FEO^{”}=\frac{22 \sqrt{34}-11}{66}×\frac{2 \sqrt{2}}{3}≒1.6754.$$
Since the length of the rectangle $ABCD$ is $GH=GO+FO^{”}+O^{”}L+JC$,
$$GH≒4+1.6754+3√2+3≒12.9180.$$
Next, regarding $∆OO’P$,
$$OO’=7,$$
$$PO=O^{”}F+O^{”}L≒1.6784+3√2≒5.9210,$$
$$O’P^2=OO’^2-PO^2≒7^2-5.9210^2=13.941759,$$
$$∴ \quad O’P≒3.7339.$$
Since the width is $IJ=IP+O’ P+O’ J$,
$$IJ=IP+O’P+O’J≒4+3.7339+3=10.7339.$$
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$$Ans. \quad Length: \ 12.9180 \ ins.; \quad Width: \ 10.7339 \ ins.$$
Reference
Yoshikazu Yamakawa, ed. (1997) Okayama ken no Sangaku (Sangaku in Okayama Prefecture), p.53; p.297-298.