I am given $3$ points $(x_i,y_i,z_i) \in \mathbb{R}^3$, for $i=1,2,3$, satisfying the following two polynomial equations:

$3+z_1+z_2+z_3-1/4(z_1x_2x_3 + z_2x_3x_1 + z_3x_1x_2) - 1/4(z_1y_2y_3 + z_2y_3y_1 + z_3y_1y_2) - 1/2(x_1x_2+x_2x_3+x_3x_1) - 1/2(y_1y_2+y_2y_3+y_3y_1) = 0$

and

$z_1(x_2y_3-x_3y_2) + z_2(x_3y_1-x_1y_3) + z_3(x_1y_2-x_2y_1) = 0$

Notice that the two polynomial equations are invariant when permuting the $3$ points in $\mathbb{R}^3$.

Prove that there does not exist $c_i > 0$, such that

$\mathbf{0} = \sum_{i=1}^3 c_i \mathbf{x}_i$

where $\mathbf{x}_i = (x_i,y_i,z_i)$ ($1 \leq i \leq 3$). Any ideas anyone?