Question 78
Suppose $a$, $b$, $c$ are the side lengths of a triangle $ABC$. Prove that
$$
2ab \sin \left(\frac{C}{2}\right) + 2bc \sin \left(\frac{A}{2}\right) + 2ca \sin \left(\frac{B}{2}\right) \le a^2 + b^2 + c^2;
$$
and that the inequality is strict unless the triangle $ABC$ is equilateral.
$$
2ab \sin \left(\frac{C}{2}\right) + 2bc \sin \left(\frac{A}{2}\right) + 2ca \sin \left(\frac{B}{2}\right) \le a^2 + b^2 + c^2;
$$
and that the inequality is strict unless the triangle $ABC$ is equilateral.