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# In a triangle $\bigtriangleup ABC$,a,b,c are its sides ,A,B $\xi\;$ C are its angles, if $\cot A,\cot B\;\xi\;\cot C$ are in AP , the a,b, and c are in which progression?

$(a)\;AP\qquad(b)\;GP\qquad(c)\;HP\qquad(d)\;No\;such\;relationship\;exists$

Explanation : $\cot A,\cot B \;\xi\;\cot C$ are in AP
$2 \cot B=\cot A+\cot C$
$2\sqrt\frac{(s(s-b))}{(s-a)(s-c)}=\sqrt\frac{s(s-a)}{(s-b)(s-c)}+\sqrt\frac{s(s-c)}{(s-a)(s-b)}$
$2(s-b)=(s-a)+(s-c)$
$2b=a+c$
$a,b\;\xi\;c\;are\;in\;AP.$