Let
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
be an arbitrary triangle and
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
a point inside it. Let
![d_a, d_b, d_c](/media/m/2/b/4/2b424afbf39599fbb1f931b961fb9be4.png)
be the distances from
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
to sides
![BC,CA,AB](/media/m/9/8/c/98c204ffa459114826231180fce7ec09.png)
;
![a, b, c](/media/m/9/e/9/9e9dfe78930065fbe5a777e9b07c27c4.png)
the lengths of the sides respectively, and
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
the area of the triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
. Prove the inequality
![abd_ad_b + bcd_bd_c + cad_cd_a \leq \frac{4S^2}{3}.](/media/m/b/d/3/bd3b82df90f1216a7b8dba0a14207b87.png)
Prove that the left-hand side attains its maximum when
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
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.