Question

In $\Delta$ $ABC$ the sides opposite to angles $A,B,C$ are denoted by $a,b,c$ respectively.

If $\cot \frac{A}{2},\cot \frac{B}{2},\cot \frac{C}{2}$ are in A.P. then $a,b,c$ are in?

• A. $A.P.$
• B. $G.P.$
• C. $H.P.$
• D. none of these

Answer 1

The correct option is A $A.P.$

if $cot\frac{A}{2},cot\frac{B}{2},cot\frac{C}{2}$ are in A.P.

then

$cot\frac{A}{2}+cot\frac{C}{2}=2cot\frac{B}{2}\Rightarrow \frac{s(s-a)+s(s-c)}{△}=\frac{2s(s-b)}{△}\Rightarrow 2s^2-sa-sc=2s^2-2sb$ $\Rightarrow sa+sc=2sb\Rightarrow a+c=2b$

Hence, $a,b,c$ are in A.P.