 # Understand the problem

Let $a, b, c$ be the lengths of sides of a (possibly degenerate) triangle. Prove the inequality $(a+b)\sqrt{ab}+(a+c)\sqrt{ac}+(b+c)\sqrt{bc} \geq (a+b+c)^2/2.$

Let $a, b, c$ be the lengths of sides of a triangle. Prove the inequality $(a+b)\sqrt{ab}+(a+c)\sqrt{ac}+(b+c)\sqrt{bc} \geq (a+b+c)^2/2.$

Inequalities
Easy
##### Suggested Book
An Excursion in Mathematics

Do you really need a hint? Try it first!

The inequality is homogeneous, hence it might be a good idea to make the substitutions $a=2R\sin A$ etc. The triangle inequality might also come in handy.
As the triangle is allowed to be degenerate, we have $a+b\ge c$. This implies $ac+bc\ge c^2$.
Summing, we get $2(ab+bc+ca)\ge a^2+b^2+c^2$. It is possible to use the AM-GM inequality here to get the LHS of the inequality. Also, show that this implies $2(ab+bc+ca)\ge\frac{(a+b+c)^2}{2}$.
Note that $(a+b)\sqrt{ab}\ge 2ab$ by AM-GM. Hence, we get $LHS\ge 2(ab+bc+ca)\ge a^2+b^2+c^2\ge\frac{(a+b+c)^2}{2}$.

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