Problem
If the differences between the hypotenuse $CA$ and the short side $AB$ of a right-angled triangle and between the hypotenuse $CA$ and the long side $BC$ are $a$ and $b$ respectively, what are the lengths of the short side, the long side, and the hypotenuse, and the diameter $d$ of the circle inscribed in the three sides? ?
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Solution
Firstly, we know that
$$CA-AB=a, —–[1]$$
$$CA-BC=b, —–[2]$$
$$d=AB+BC-CA. —–[3]$$
Squaring both sides of [3] gives
$$d^2=AB^2+BC^2+CA^2+2AB\cdot BC-2AB\cdot CA-2BC\cdot CA.$$
Because $AB^2+BC^2=CA^2,$
$$\begin{eqnarray}&&d^2=2CA^2-2AB\cdot CA-2BC\cdot CA+2AB\cdot BC\\[5pt]&=&2CA(CA-AB)-2BC(CA-AB) =2(CA-AB)(CA-BC) \\[5pt]&=&2ab,\end{eqnarray}$$
$$∴d=\sqrt{2ab}.$$
From [1]+[2], we have
$$2CA-AB-BC=a+b,$$
$$CA+(CA-AB-BC)=a+b,$$
$$CA=a+b+(AB+BC-CA),$$
$$∴ CA=a+b+d=a+b+\sqrt{2ab},$$
($∵ d=AB+BC-CA$)
$$∴ AB=CA-a=b+\sqrt{2ab},$$
$$∴ BC=CA-b=a+\sqrt{2ab}.$$
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$$Ans.\quad AB=b+\sqrt{2ab},$$
$$BC=a+\sqrt{2ab},$$
$$CA=a+b+\sqrt{2ab},$$
$$d=\sqrt{2ab}.$$
Reference
Yoshikazu Yamakawa, ed. (1997) Okayama ken no Sangaku (Sangaku in Okayama Prefecture), p.51; pp.308-309.