Question

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

Answer 1

In $\Delta$ABC,

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

$\cot \frac{A}{2}=\sqrt{\frac{s(s-a)}{(s-b)(s-c)}}$

$\cot \frac{B}{2}=\sqrt{\frac{s(s-b)}{(s-a)(s-c)}}$

$\cot \frac{C}{2}=\sqrt{\frac{s(s-c)}{(s-a)(s-b)}}$

$2\cot \frac{B}{2}=\cot \frac{A}{2}+\cot \frac{C}{2}$

$2\sqrt{\frac{\text{/}s(s-b)}{(s-a)(s-c)}}=\sqrt{\frac{\text{/}s(s-a)}{(s-b)(s-c)}}+\sqrt{\frac{\text{/}s(s-c)}{(s-a)(s-b)}}$

$2\sqrt{(s-b{)}^2}=\sqrt{(s-a{)}^2}+\sqrt{(s-c{)}^2}$

$\text{/}2s-2b=\text{/}s-a+\text{/}s-c$

$2b=a+c$

So, a, b, c are in A.P.