Question 51
Let $a = BC$, $b = CA$ and $c = AB$ be respectively the lengths of a triangle $ABC$; $i_a, i_b, i_c$ be respectively the lengths of the angle bisectors from $A, B$ and $C$. Let $R$ be the circumradius of the triangle. Prove that
$$
ai_a + bi_b + ci_c \lt 9R^2.
$$
$$
ai_a + bi_b + ci_c \lt 9R^2.
$$