In a triangle $A B C, tan \frac{A}{2}, tan \frac{B}{2}, tan \frac{C}{2}$ are in harmonic progression, then show that the sides $a, b, c$ are in Arithmetic progression.

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Solution

In a triangle $A B C, tan \frac{A}{2}, tan \frac{B}{2}, tan \frac{C}{2}$ are in H.P
$\Rightarrow, cot \frac{A}{2}, cot \frac{B}{2}, cot \frac{C}{2}$ are in A.P
$\Rightarrow 2 cot \frac{B}{2} = cot \frac{A}{2} + cot \frac{C}{2}$
$\Rightarrow 2 \sqrt{\frac{s (s - b)}{(s - a) (s - c)}} = \sqrt{\frac{s (s - a)}{(s - b) (s - c)}} + \sqrt{\frac{s (s - c)}{(s - a) (s - b)}}$
Multilply through out by $\sqrt{s (s - a) (s - b) (s - c)}$
$\Rightarrow 2 (s - b) = (s - a) + (s - c)$
$\Rightarrow 2 s - 2 b = 2 s - a - c$
$\Rightarrow 2 b = a + c$
$∴ a, b, c$ are in A.P

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