Problem
There are three circles in a right triangle $ABC$ as shown in the figure.
When the product of the diameter of the large circle and the length of $DA$ is 4.8 and the difference between the diameter of the middle circle and the side $AB$ is 1.4 inches, what is the diameter of the middle circle?
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$\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$
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Solution
Let $l$ be the diameter of the large circle, and $y$ be the diameter of the medium circle.
And also let:
$DA×l=p —[1],$ and $AB-y=k —[2].$
From [1] and [2], it follows that
$l=\frac{p}{DA} —[1]’,$ and $AB=k+y —[2]’.$
From $∆ABC∼∆DAC$, we have
$$\frac{l}{AB}=\frac{y}{DA},$$
$$\frac{\frac{p}{DA}}{k+y}=\frac{y}{DA},$$
$$p=y×(k+y),$$
$$y^2+ky-p=0,$$
$$y=\frac{-k±\sqrt{k^2+4p}}{2},$$
$$\therefore y=\frac{\sqrt{k^2+4p}-k}{2} —[3],$$
$(\because \frac{-\sqrt{k^2+4p}+k}{2}<0).$
Since $p = 4.8, k = 1.4$ from the problem statement, we see that
$$y=\frac{\sqrt{1.4^2+4×4.8}-1.4}{2}=1.6.$$
(Answer) 1.6 inches.
Reference
Yoshikazu Yamakawa, ed. (1997) Okayama ken no Sangaku (Sangaku in Okayama Prefecture), p.55; pp.292-294.
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