Proving a determinant equality 
Prove
$$\begin{vmatrix}
        2bc-a^2 & c^2 & b^2 \\
        c^2 & 2ac-b^2   & b^2 \\
        b^2 & a^2 & 2ab-c^2\\
        \end{vmatrix}
=(a^3+b^3+c^3-3abc)^2$$

My attempt:
I tried using the well-known result that 
$$\begin{vmatrix}
        bc-a^2 & ca-b^2 & ba-c^2 \\
        ac-b^2& ab-c^2   & bc-a^2 \\
        ba-c^2& bc-a^2 & ca-b^2\\
        \end{vmatrix}
=\begin{vmatrix}
        a & b & c \\
        b& c   & a \\
        c& a & b\\
        \end{vmatrix}^2
=(a^3+b^3+c^3-3abc)^2$$
But I tried using many properties of determinant but I was unable to bring the l.h.s. to the form $$\begin{vmatrix}
        bc-a^2 & ca-b^2 & ba-c^2 \\
        ac-b^2& ab-c^2   & bc-a^2 \\
        ba-c^2& bc-a^2 & ca-b^2\\
        \end{vmatrix}$$
Please help this one out. Thanks.

Hint: Observe
\begin{align}
\begin{bmatrix}
2bc-a^2 & c^2 & b^2\\
c^2 & 2ac-b^2 & a^2\\
b^2 & a^2 & 2ab-c^2
\end{bmatrix}
=
\begin{bmatrix}
a & b & c\\
b & c & a\\
c & a & b
\end{bmatrix}
\begin{bmatrix}
-a & -b & -c\\
c & a & b\\
b & c & a\\
\end{bmatrix}
\end{align}
Edit: Just take the determinant.
