Question 31
Let $ABC$ be an acute triangle and $P$ be a point inside the triangle such that $\angle APB = \angle BPC = \angle CPA$. Denote with $S$ the area and with $\alpha, \beta, \gamma$ the angles of $\triangle ABC$. Prove that
$$
\frac{1}{\sin \alpha} + \frac{1}{\sin \beta} + \frac{1}{\sin \gamma} \ge \frac{PA^2 + PB^2 + PC^2}{2S} + \frac{4}{\sqrt{3}}
$$
When does the equality occur?
$$
\frac{1}{\sin \alpha} + \frac{1}{\sin \beta} + \frac{1}{\sin \gamma} \ge \frac{PA^2 + PB^2 + PC^2}{2S} + \frac{4}{\sqrt{3}}
$$
When does the equality occur?