<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] First apply AM-GM inequality: $(a+b+c)^3 \geq 27abc$, then you need to prove: $abc \geq (a+b-c)(b+c-a)(c+a-b)$, and this follows from:
$a^2 \geq a^2 - (b-c)^2$
$b^2 \geq b^2 - (c-a)^2$
$c^2 \geq c^2 - (a-b)^2$.
by multiply all three inequalities and taking square root.
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