Question 57
The triangle $ABC$ has the semiperimeter $p$, the circumradius $R$, the inradius $r$ and $l_a, l_b, l_c$ are the lengths of internal bisectors, drawn from the vertices $A, B$ and $C$ respectively. Prove the inequality
$$
l_a l_b + l_b l_c + l_c l_a \leq p \sqrt{3 r^2 + 12 R r}.
$$
$$
l_a l_b + l_b l_c + l_c l_a \leq p \sqrt{3 r^2 + 12 R r}.
$$