Question 52
Let $w_a, w_b, w_c$ be the lengths of the internal angle bisectors of a triangle $ABC$ with sides $a, b, c$.
Let $R$ be its circum-radius. Prove that
$$
\frac{b^2 + c^2}{w_a} + \frac{c^2 + a^2}{w_b} + \frac{a^2 + b^2}{w_c} \gt 4R.
$$
Let $R$ be its circum-radius. Prove that
$$
\frac{b^2 + c^2}{w_a} + \frac{c^2 + a^2}{w_b} + \frac{a^2 + b^2}{w_c} \gt 4R.
$$