$(x+y+z)^3-(y+z-x)^3-(z+x-y)^3-(x+y-z)^3=24xyz$? The question given is 

Show that $(x+y+z)^3-(y+z-x)^3-(z+x-y)^3-(x+y-z)^3=24xyz$.

What I tried is suppose $a=(y+z-x),\ b=(z+x-y)$ and $c=(x+y-z)$ and then noted that $a+b+c=x+y+z$. So the question statement reduced to $(a+b+c)^3-(a^3+b^3+c^3)$. Then I tried to invoke the identity $(a^3+b^3+c^3-3abc)=(a+b+c)(a^2+b^2+c^2-ab-bc-ac)$ by adding and subtracting $3abc$ in the question statement. After doing all this when I substituted back the values of $a,b$ and $c$, I ended up with the initial question statement.
Any hints will be appreciated.

Observe that,
$\begin{align}(x+y+z)^3-(x+y-z)^3&=2z\left((x+y+z)^2+(x+y-z)^2)+(x+y)^2-z^2\right)\\&=2z\left(3(x+y)^2+z^2\right)\tag{1}\end{align}$
and,
$\begin{align}(x-y-z)^3-(x-y+z)^3&=-2z\left((x-y-z)^2+(x-y+z)^2)+(x-y)^2-z^2\right)\\&=2z\left(3(x-y)^2+z^2\right)\tag{2}\end{align}$
Therefore,
$(x+y+z)^3+(x-y-z)^3-(x-y+z)^3-(x+y-z)^3\\=\bigl((x+y+z)^3-(x+y-z)^3\bigr)+\bigl((x-y-z)^3-(x-y+z)^3\bigr)\\=2z\left(3(x+y)^2+z^2\right)-2z\left(3(x-y)^2+z^2\right)\\=2z(3\cdot 4xy)\\=24xy$
