If the cotangents of half the angles of a triangle are in A.P., then the sides are in :

A.P.

$\cot\frac{A}{2}=\sqrt{\frac{s(s-a)}{(s-b)(s-b)}}=\frac{s(s-a)}{Δ};\cot\frac{B}{2}=\frac{s(s-b)}{Δ},\cot\frac{C}{2}=\frac{s(s-c)}{Δ}$

Given $\cot\frac{B}{2},\cot\frac{B}{2},\cot\frac{C}{2}$ are in A.P. ⇒ $(s-a),(s-b),(s-c)$ are in A.P. ⇒ a, b, c are in A.P.