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lf the length...

Question

lf the length of tangents from vertices to Incircle are in $H . P$ , then $r_{1}, r_{2}, r_{3}$ are in

A.P

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G.P

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H.P

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A.G.P

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Solution

The correct option is A A.P
Lengths of the triangles are in H.P then,
From the relation $S = r_{a} (p - a)$
So, $\frac{1}{(p - a)} = \frac{r_{a}}{S}$
and other similar relations

From question
$\frac{1}{p - a} + \frac{1}{p - c} = 2 (\frac{1}{p - b})$
$\frac{r_{a}}{S} + \frac{r_{c}}{S} = 2 (\frac{r_{b}}{S})$
Now, required expression is
$r_{a} + r_{c} = 2 r_{b}$
Hence, these are in A.P

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