Question 90
Let $ABC$ be a triangle with side-lengths $a$, $b$, $c$, inscribed in a circle with radius $R$ and let $I$ be its incenter. Let $P_{1}$, $P_{2}$ and $P_{3}$ be the areas of the triangles $ABI$, $BCI$ and $CAI$, respectively. Prove that
$$
\frac{R^{4}}{P_{1}^{2}}+\frac{R^{4}}{P_{2}^{2}}+\frac{R^{4}}{P_{3}^{2}} \geq 16
$$
$$
\frac{R^{4}}{P_{1}^{2}}+\frac{R^{4}}{P_{2}^{2}}+\frac{R^{4}}{P_{3}^{2}} \geq 16
$$