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Combination

Prove r1 -...

Question

Prove $(r_{1} - r) / a + (r_{2} - r) / b = c / r_{3}$

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Solution

$\Rightarrow \frac{r_{1} - r}{a} + \frac{r_{2} - r}{b} = \frac{\frac{S}{s - a} - \frac{S}{s}}{a} + \frac{\frac{S}{s - b} - \frac{S}{s}}{b}$
$= \frac{S [s - (s - a)]}{s (s - a) a} + \frac{S [s - (s - b)]}{s (s - b)}$
$= \frac{S}{s (s - a)} + \frac{S}{s (s - b)}$
$= \frac{S}{s} [\frac{1}{s - a} + \frac{1}{s - b}]$
$= \frac{S}{s} [\frac{(s - a + s + b)}{(s - a) (s - b)}]$
$= \frac{S [C]}{s (s - a) (s - b)}$
$= \frac{S C}{\frac{S^{2}}{s - C}}$
$= \frac{(s - c)}{S} C = \frac{C}{r_{3}}$ proved.

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