A counterexample is $c_1= 2,c_2= 2,c_3= 81,x_1= -8,x_2= -1,x_3= \frac{2}{9},y_1= 0,y_2= 0,y_3= 0$, $z_1= 0,z_2= 0,z_3= 0$.

A counterexample is $c_1= 2,c_2= 2,c_3= 81,x_1= -8,x_2= -1,x_3= \frac{2}{9},y_1= 0,y_2= 0,y_3= 0$, $z_1= 0,z_2= 0,z_3= 0$.

Added in response to the OP's comment: With the additional condition that the points $\mathbf x_i=(x_i,y_i,z_i)$ be nonzero, we still have a counterexample: $c_1= 1,c_2= 1,c_3= \frac{5233}{128},x_1= -8,x_2= -1,x_3= \frac{1152}{5233},y_1= \frac{7}{8},y_2= 0,y_3= -\frac{112}{5233}$, $z_1= 0,z_2= 0,z_3= 0$.