By elimination of $a,b,c$from the homogeneous equations  $x=\frac{a}{b-c},y=\frac{b}{c-a},z=\frac{c}{a-b}$

where $a,b,c$not all zero. 

$a-xb+xc=0\to \left(1\right)ya+b-yc=0\to \left(2\right)$

$za-zb-c=0\to \left(3\right)$

Since a, b, c are not all zero, the above system has a non-trivial solution.

By eliminating $a,b,c$from $\left(1\right),\left(2\right),\left(3\right),$we get

$\left|\begin{array}{ccc}1& -x& x\\ y& 1& -y\\ z& -z& -1\end{array}\right|=0\Rightarrow xy+yz+zx=-1$