while seeking to devise an intuitive, computation-lite demonstration of the fact that inversion preserves the concyclicity or collinearity of four points in the complex plane, i hit on the following idea, which initially seemed promising:
let $z_1,z_2,z_3,z_4$ be four distinct complex numbers, none of which is zero.
if we define the determinant: $$ M=M(z_1,z_2,z_3,z_4)= \begin{vmatrix} |z_1|^2 & \mathfrak{Re} z_1 & \mathfrak{Im}z_1 & 1 \\ |z_2|^2 & \mathfrak{Re} z_2 & \mathfrak{Im}z_2 & 1 \\ |z_3|^2 & \mathfrak{Re} z_3 & \mathfrak{Im}z_3 & 1 \\ |z_4|^2 & \mathfrak{Re} z_4 & \mathfrak{Im}z_4 & 1 \\ \end{vmatrix} $$ then, since $|z^{-1}| = |z|^{-1}$, and $\mathfrak{Re} z^{-1}=\frac{\mathfrak{Re}z}{|z|^2},\mathfrak{Im} z^{-1}=-\frac{\mathfrak{Im}}{|z|^2}$ we have $$ M^*=M(z_1^{-1},z_2^{-1},z_3^{-1},z_4^{-1}) = \frac{M(z_1,z_2,z_3,z_4)}{|z_1|^2 |z_2|^2|z_3|^2|z_4|^2} $$ i.e. $M=0$ if and only if $M^*=0$
(i think) it is straightforward to show that if the four $z_i$ are concyclic or collinear, then the corresponding determinant is zero.
however, for this idea to have traction, the reverse implication is required, i.e. that if a determinant of this type is zero, then the four points are either concyclic or collinear.
question can anyone help? either by pointing out some glitch in my reasoning, or by supplying an argument for the reverse implication required to develop the proof-idea?
