2017/030)In a triangle if $\cot\,A\,\cot\,B\,\cot\,C$ are in AP then $a^2,b^2,c^2$ are in $\cdots$ progression (IIT1985-5 marks)

We have $\cot\,A\,\cot\,B\,\cot\,C$  are in AP
so
$\cot\, A + \cot\, C = 2\cot\, B$
or $\frac{\cos\, A }{\sin\, A }  +\frac{\cos\, C }{\sin\, C } = \frac{\cos\, B }{\sin\, B }$
or $\frac{\cos\, A \sin\, C + \sin\, A  \cos\, C}{\sin\, A \sin\, C } = \frac{\cos\, B }{\sin\, B} $
or $\frac{\sin ( A + C) }{\sin\, A \sin\, C }  = \frac{\cos\, B }{\sin\, B} $
or  $\frac{\sin B}{\sin\, A \sin\, C }  = \frac{\cos\, B }{\sin\, B} $ as in a triangle $\sin (A+B) = \sin\,c$
or  $\sin^2 B = 2\sin\, A \sin\, C\cos \,B $
using law of sin an cos we get
$b^2 = 2ac\frac{a^2+c^2-b^2}{ac}$
or $b^2 = a^2+c^2-b^2$
or $2b^2 = a^2 +c ^2$
hence $a^2,b^2,c^2$ are in AP