Prove determinant is negative Prove that the determinant $\Delta$ is negative

$$
\Delta=\begin{vmatrix}
a & b & c \\
b & c & a \\
c & a & b
\end{vmatrix}<0
$$

where $a,b,c$ are positive and $a\neq b\neq c$.
My Attempt:
Applying Sarrus' rule,
$$
\begin{matrix}
a&b&c&a&b\\
b&c&a&b&c\\
c&a&b&c&a
\end{matrix}
$$
$$
\Delta=acb+bac+cba-c^3-a^3-b^3=3abc-(a^3+b^3+c^3)\\
=-\Big[(a^3+b^3+c^3)-3abc\Big]
$$
How do I prove that $(a^3+b^3+c^3)-3abc>0$ thus prove $\Delta<0$ ?

By the AM-GM inequality:
$$
\frac{a^3+b^3+c^3}{3} \ge \sqrt[3]{a^3b^3c^3} = abc
$$
The strict inequality holds unless $\,a=b=c\,$.
