In a triangle ABC if cotA,cotB,cotC are in A.P., thenn a,b,c are in progression.

n cot A, cot B, cot C are in A.P n => cot B=(cot A+cot C / 2) n => 2 cot B=cot A+cot C n => 2 sqrt((s(s-b) / (s-a)(s-c)))=sqrt((s(s-a) / (s-b)(s-c)))+sqrt((s(s-c) / (s-a)(s-b))) n => 2(s-b)=(s-a)+(s-c) => 2 b=a+c n => a, b, c are in A.P. n\

In a triangle ABC if cotA,cotB,cotC are in A.P., then ... | Sequence & Series (Mathematics)

\begin{aligned}\n& \cot A, \cot B, \cot C \text { are in A.P } \\\n& \Rightarrow \cot B=\frac{\cot A+\cot C}{2} \\\n& \Rightarrow 2 \cot B=\cot A+\cot C \\\n& \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)}} \\\n& \Rightarrow 2(s-b)=(s-a)+(s-c) \Rightarrow 2 b=a+c \\\n& \Rightarrow a, b, c \text { are in A.P. }\n\end{aligned}

Explanation

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