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Cartesian Product

The exradii r...

Question

The exradii $r_{1}, r_{2} a n d r_{3}$ of a $△$ ABC are H.P , then its sides a, b, c are in

A

GP

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B

HP

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C

AP

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D

None of these

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Solution

The correct option is C
AP

$r_{1}, r_{2} a n d r_{3}$ are in H.P $\Rightarrow \frac{2}{r_{2}} = \frac{1}{R_{1}} + \frac{1}{r_{3}}$
$\Rightarrow \frac{2 (s - b)}{△} = \frac{(s - a)}{△} + \frac{(s - c)}{△}$
2s-2b=2s-a-c
$\Rightarrow 2 b = a + c$ . Hence a,b,c are in A.P

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