Let
be an arbitrary triangle and
a point inside it. Let
be the distances from
to sides
;
the lengths of the sides respectively, and
the area of the triangle
. Prove the inequality
Prove that the left-hand side attains its maximum when
is the centroid of the triangle.
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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
$$abd_ad_b + bcd_bd_c + cad_cd_a \leq \frac{4S^2}{3}.$$
Prove that the left-hand side attains its maximum when $M$ is the centroid of the triangle.
Školjka