complex number with determinant Let $z_1$ and $z_2$ be two distinct complex numbers and let
$\,z=(1-t)z_1 +tz_2\,$ for some real number $t$ with $0<t<1$. Then we have to prove 
$$\begin{vmatrix} z-z_1 & \overline{z}-\overline{z_1} \\
z_2-z_1 & \overline{z_2}-\overline{z_1} 
\end{vmatrix}\;=\;0$$
I thought about it, but don't get any start.
Can anybody provide me a hint?
can anybody help me in this

Given that $z=(1-t)z_1+tz_2$ we can write
$z=z_1+t(z_2-z_1)$ so that $z-z_1=t(z_2-z_1)$ and
$$\overline{z}-\overline{z_1}=\overline{z-z_1}=\overline{t(z_2-z_1)}=t(\overline{z_2}-\overline{z_1}),$$
so the top row of the matrix is $t$ times the bottom row. Hence the determinant is zero.
