Question 3
Let $ABC$ be a triangle inscribed in a circle. Put $l_a=m_a/M_a$, $l_b=m_b/M_b$, $l_c=m_c/M_c$, where $m_a,m_b,m_c$ are the internal angle-bisector lengths and $M_a,M_b,M_c$ are the lengths of these bisectors extended to the circle. Prove that
$$\frac{l_a}{\sin^2A}+\frac{l_b}{\sin^2B}+\frac{l_c}{\sin^2C}\ge3,$$
with equality if and only if $ABC$ is equilateral.
In the constructed diagram $D_A,D_B,D_C$ are the intersections with the opposite sides and $T_A,T_B,T_C$ the second intersections with the circle.
$$\frac{l_a}{\sin^2A}+\frac{l_b}{\sin^2B}+\frac{l_c}{\sin^2C}\ge3,$$
with equality if and only if $ABC$ is equilateral.
In the constructed diagram $D_A,D_B,D_C$ are the intersections with the opposite sides and $T_A,T_B,T_C$ the second intersections with the circle.