Here $z_1,z_2,z_3,z_4$ are distinct complex numbers, $k$ a complex variable.
I want to show that the following system of linear equations in variables $a,b,c,d$ has a nontrivial solution.
$az_1 + b -cz_1 -d = 0$
$az_2 + b +cz_2 +d = 0$
$az_3 + b -kcz_3 -kd = 0$
$az_4 + b +kcz_4 +kd = 0$
I tried putting the coefficients in a matrix, computing the determinant, and simplifying to get:
$2k((z_3 - z_2)(z_1-z_4) + (z_3 - z_1)(z_2 - z_4)) + (k^2 + 1)(z_4 - z_3)(z_1 - z_2).$
But i'm not sure how to show this quantity is nonzero.