Question

In triangle ABC, prove that the maximum value of
$\tan \frac{A}{2}\tan \frac{B}{2}\tan \frac{C}{2}$ is $\frac{R}{2s}$

Answer 1

For triangle ABC, we have,
$\tan \frac{A}{2}\tan \frac{B}{2}\tan \frac{C}{2}=\sqrt{\frac{(s-b)(s-c)}{s(s-a)}}\sqrt{\frac{(s-a)(s-c)}{s(s-b)}}\sqrt{\frac{(s-a)(s-b)}{s(s-c)}}$
$=\frac{\sqrt{s(s-a)(s-b)(s-c)}}{s^2}$
$=\frac{\Delta }{s^2}=\frac{r}{s}\le \frac{R}{2s}$