Question

In $\Delta ABC$, if $3\tan \frac{A}{2}\tan \frac{C}{2}=1$, then $a,b,c$ are in:

• A. $A.P$
• B. $G.P$
• C. $H.P$
• D. $A.G.P$

Answer 1

The correct option is A $A.P$

We know, $\tan \frac{A}{2}=\sqrt{\frac{(s-b)(s-c)}{s(s-a)}}$
$\therefore 3\tan \frac{A}{2}\tan \frac{C}{2}=3\sqrt{\frac{(s-b)(s-c)}{s(s-a)}}\cdot \sqrt{\frac{(s-a)(s-b)}{s(s-c)}}$
$\Rightarrow 3\tan \frac{A}{2}\tan \frac{C}{2}=3\times \frac{s-b}{s}=3\times \frac{a-b+c}{a+b+c}=1$
$\Rightarrow 3a-3b+3c=a+b+c$
$\Rightarrow 2a+2c=4b$
$\Rightarrow a+c=2b$
Hence $a,b,c$ are in $A.P.$