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If in a ABC...

Question

If in a $△ A B C, r_{1} r_{2} + r_{2} r_{3} + r_{3} r_{1}$ is equal to
(where $r_{1}, r_{2}$ and $r_{3}$ are the exradii and $2 s$ is the perimeter.)

s^{2}

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2 s^{2}

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3 s^{2}

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4 s^{2}

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Solution

The correct option is A $s^{2}$
$r_{1} r_{2} + r_{2} r_{3} + r_{3} r_{1}$
$= \frac{△}{s - a} \frac{△}{s - b} + \frac{△}{s - b} \frac{△}{s - c} + \frac{△}{s - c} \frac{△}{s - a}$
$= △^{2} [\frac{1}{(s - a) (s - b)} + \frac{1}{(s - b) (s - c)} + \frac{1}{(s - c) (s - a)}]$
By Heron's formula
$s (s - a) (s - b) (s - c) [\frac{1}{(s - a) (s - b)} + \frac{1}{(s - b) (s - c)} + \frac{1}{(s - c) (s - a)}]$
$= [\frac{s (s - a) (s - b) (s - c)}{(s - a) (s - b)} + \frac{s (s - a) (s - b) (s - c)}{(s - b) (s - c)} + \frac{s (s - a) (s - b) (s - c)}{(s - c) (s - a)}]$
$= s (s - c) + s (s - a) + s (s - b)$
$= s (3 s - a - b - c)$
$= s (3 s - 2 s)$
$= s^{2}$

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