$1 / r_{1} + 1 / r_{2} + 1 / r_{3} = 1 / r$

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Solution

To prove: $\frac{1}{r_{1}} + \frac{1}{r_{2}} + \frac{1}{r_{3}} = \frac{1}{r}$
$\Rightarrow \frac{1}{r_{1}} + \frac{1}{r_{2}} + \frac{1}{r_{3}} = \frac{s - a}{S} + \frac{s - b}{S} + \frac{s - c}{S}$
$= \frac{3 s - (a + b + c)}{S}$
We know, $s = \frac{a + b + c}{2} \Rightarrow a + b + c = 2 s$
$\Rightarrow \frac{1}{r_{1}} + \frac{1}{r_{2}} + \frac{1}{r_{3}} = \frac{3 r - 2 r}{S} = \frac{r}{S} = \frac{1}{r} =$ R.H.S proved.

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