APMO 1997 Problem 3

Let $ABC$ be a triangle inscribed in a circle and let $$l_a = \frac{m_a}{M_a} \ , \ \ l_b = \frac{m_b}{M_b} \ , \ \ l_c = \frac{m_c}{M_c} \ ,$$ where $m_a$ , $m_b$ , $m_c$ are the lengths of the angle bisectors (internal to the triangle) and $M_a$ , $M_b$ , $M_c$ are the lengths of the angle bisectors extended until they meet the circle. Prove that $$\frac{l_a}{\sin^2 A} + \frac{l_b}{\sin^2 B} + \frac{l_c}{\sin^2 C} \geq 3$$ and that equality holds iff $ABC$ is an equilateral triangle.