<s>[INST] Calculate the inequality $a,b,c$ are the sides of a triangle.Then show that
$$(a+b+c)^3 > 27(a+b-c)(b+c-a)(c+a-b)$$ Also give the case where equality holds i.e. $$(a+b+c)^3=27(a+b-c)(b+c-a)(c+a-b)$$
I tried triangle inequality, AM-GM inequality in many way but cannot help my cause. 
 [/INST] By AM-GM 
$$(a+b+c)^3=\left(\sum_{cyc}a\right)^3=\left(\sum_{cyc}(a+b-c)\right)^3\geq$$
$$\geq\left(3\sqrt[3]{\prod_{cyc}(a+b-c)}\right)^3=27(a+b-c)(a+c-b)(b+c-a).$$
Done!
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