Prove that: `|(b+c)^2a^2a^2b^2(c+a)^2b^2c^2c^2(a+b)^2|=2a b c(a+b+c)^3`

Answer

Proved

Explanation

$$\Delta=\left|\begin{array}{ccr}a-b-c & 2 a & 2 a \\ 2 b & b-c-a & 2 b \\ 2 c & 2 c & c-a-b\end{array}\right|$$


Applying `R_{1} \rightarrow R_{1}+R_{2}+R_{3}`, we have:
$$
\Delta =\left|\begin{array}{lll}
a+b+c & a+b+c & a+b+c \\
2 b & b-c-a & 2 b \\
2 c & 2 c & c-a-b
\end{array}\right| \\
...