Question

If the sides $a,b,c$ of $△ABC$ are in $A.P.$, prove that $\cot \frac{1}{2}A,\cot \frac{1}{2}B,\cot \frac{1}{2}C$ are in $A.P.$

Answer 1

$\cot \frac{1}{2}A,\cot \frac{1}{2}B,\cot \frac{1}{2}C$ are in $A.P.$

If $2\cot \frac{1}{2}B=\cot \frac{1}{2}A+\cot \frac{1}{2}C$

i.e. if $2\sqrt{\left\{\frac{s(s-b)}{(s-c)(s-a)}\right\}}$

=$\sqrt{\left\{\frac{s(s-a)}{(s-b)(s-c)}\right\}}+\sqrt{\left\{\frac{s(s-c)}{(s-a)(s-b)}\right\}}$

i.e. if $2(s-b)=(s-a)+(s-c)$

i.e. if $2b=a+c$

i.e. if $a,b,c$ are in $A.P.$