In any $\mathrm{△}ABC,$if $\cot \frac{A}{2},\cot \frac{B}{2},\cot \frac{C}{2}$ are in A.P., then$a, b, c$ are in 

Solution:

We have,

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

$\begin{array}{l}\Rightarrow 2\cot \frac{B}{2}=\cot \frac{A}{2}+\cot \frac{C}{2}\\ \Rightarrow 2\sqrt{\frac{s(s-b)}{(s-a)(s-c)}}=\sqrt{\frac{s(s-a)}{(s-b)(s-c)}}+\sqrt{\frac{s(s-c)}{(s-a)(s-b)}}\end{array}$

$\Rightarrow 2(s-b)=s-a+s-c\Rightarrow 2b=a+c\Rightarrow a,b,c$ are in A.P.