Question

If ex-radii $r_1,r_2,r_3$ of a triangle are in H.P., then sides $a,b,c$ are in

• A. $\text{G.P.}$
• B. $\text{A.G.P.}$
• C. $\text{A.P.}$
• D. None of these

Answer 1

The correct option is C $\text{A.P.}$

$r_1,r_2,r_3$ is in H.P

$\frac{2}{r_2}=\frac{1}{r_1}+\frac{1}{r_3}$

$\Rightarrow \frac{2(s-b)}{\Delta }=\frac{(s-a)}{\Delta }+\frac{(s-c)}{\Delta }$

$\Rightarrow 2b=a+c$

Then, $a,b,c$ is in A.P