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 $\Delta<0$ ?

You can factor $$a^3 + b^3 + c^3 - 3 abc = (a+b+c)(a^2 + b^2 + c^2 - ab - bc - ca),$$ where the first term is certainly positive. For the second term, observe $$a^2 + b^2 + c^2 - ab-bc-ca = \frac{1}{2}((a-b)^2 + (b-c)^2 + (c-a)^2),$$ which is also positive.
