Question

In $△ABC$, if $\cot A,\cot B,\cot C$ are in A.P then show $a^2$ $,b^2,c^2$ are in A.P

Answer 1

If $\cot A,\cot B,\cot C$ are in A.P
$\Rightarrow 2\cot B=\cot A+\cot C$
$\Rightarrow 2.\frac{c^2+a^2-b^2}{4\Delta }=\frac{b^2+c^2-a^2}{4\Delta }+\frac{a^2+b^2-c^2}{4\Delta }$
$\Rightarrow 2(c^2+a^2-b^2)=2b^2\Rightarrow c^2+a^2=2b^2$
Hence $a^2,b^2,c^2\in A.P$