Question 9
(ROM 3) Let $ABC$ be an arbitrary triangle and $M$ a point inside it. Let $d_a,d_b,d_c$ be the distances from $M$ to sides $BC,CA,AB$; $a,b,c$ the lengths of the sides respectively, and $S$ the area of the triangle $ABC$. Prove the inequality
$$ab d_a d_b+bc d_b d_c+ca d_c d_a\le\frac{4S^2}{3}.$$
Prove that the left-hand side attains its maximum when $M$ is the centroid of the triangle.
$$ab d_a d_b+bc d_b d_c+ca d_c d_a\le\frac{4S^2}{3}.$$
Prove that the left-hand side attains its maximum when $M$ is the centroid of the triangle.