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A $\triangle A B C$ is right angled at $B$. Then the diameter of the incircle of the triangle is
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$r=\frac{\Delta}{s}-\frac{\left(\frac{1}{2}\right) \mathrm{ac}}{\left(\frac{1}{2}\right)(a+b+c)}$ $=\frac{a c}{(a+b+c)}=\frac{a c(c+a-b)}{(c+a)^2-b^2}=$ $\frac{a c(c+a-b)}{c^2+2 c a+a^2-b^2}$
$=\frac{a c(c+a-b)}{2 c a+b^2-b^2}=\frac{c+a-b}{2} \quad\left(\because a^2+c^2=b^2\right)$
$=\frac{a c(c+a-b)}{2 c a+b^2-b^2}=\frac{c+a-b}{2} \quad\left(\because a^2+c^2=b^2\right)$
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