Problem

If the length of the short side of a right triangle is $a$ and the diameter of the circle inscribed in the three sides is $d$, what are the lengths of the long side $x$ and the hypotenuse $y$ respectively?

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Solution

From the figure it can be seen that

$$y=a+x-d. —–[1]$$

From the Pythagorean theorem, we have

$$a^2+x^2=y^2. —–[2]$$

Substituting [1] into [2], we see that

$$a^2+x^2=(x+a-d)^2,$$

$$x=\frac{2ad-d^2}{2a-2d}. —–[3]$$

Substituting [3] into [1], it follows that

$$y=\frac{2a^2-2ad+d^2}{2a-2d}.$$

(Answer) $x=\frac{2ad-d^2}{2a-2d},$

$y=\frac{2a^2-2ad+d^2}{2a-2d}.$

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Reference

Yoshikazu Yamakawa, ed. (1997) Okayama ken no Sangaku (Sangaku in Okayama Prefecture), p.50; pp.309.