Question 93
Suppose $a$, $b$, $c$ are the side lengths of a triangle $ABC$ with area $S$, and semi-perimeter $s$. Prove that
$$
s \lt \sqrt{a^2 + b^2 - 4S} + \sqrt{b^2 + c^2 - 4S} + \sqrt{c^2 + a^2 - 4S}.
$$
$$
s \lt \sqrt{a^2 + b^2 - 4S} + \sqrt{b^2 + c^2 - 4S} + \sqrt{c^2 + a^2 - 4S}.
$$