Question 151
Show that for any triangle $ABC$ with area $S$ and circumradius $R$,
$$
(a) \quad \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} \le \frac{3\sqrt{3}}{4S} + \left(\frac{1}{a} - \frac{1}{b}\right)^2 + \left(\frac{1}{b} - \frac{1}{c}\right)^2 + \left(\frac{1}{c} - \frac{1}{a}\right)^2,
$$
$$
(b) \quad \tan \frac{A}{2} + \tan \frac{B}{2} + \tan \frac{C}{2} \le \frac{9R^2}{4S}.
$$
Here $AB = c$, $BC = a$ and $CA = b$.
$$
(a) \quad \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} \le \frac{3\sqrt{3}}{4S} + \left(\frac{1}{a} - \frac{1}{b}\right)^2 + \left(\frac{1}{b} - \frac{1}{c}\right)^2 + \left(\frac{1}{c} - \frac{1}{a}\right)^2,
$$
$$
(b) \quad \tan \frac{A}{2} + \tan \frac{B}{2} + \tan \frac{C}{2} \le \frac{9R^2}{4S}.
$$
Here $AB = c$, $BC = a$ and $CA = b$.