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
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