# If the complex numbers $z_1,\:z_2,\:z_3$ represent the vertices $A,\:B,\:C$ of a parallelogram, then the fourth vertex $D$ is ?

$\begin{array}{1 1}(A)\large\frac{1}{2}(z_1+z_2) \\ (B) z_1-z_2+z_3\\ (C)\large\frac{1}{2}(z_1+z_3) \\(D) \large\frac{1}{2}(z_2+z_3) \end{array}$

Toolbox:
• If $z_1,\:z_2$ represent two points on argand plane, then $\large\frac{z_1+z_2}{2}$ is the mid point of the segment joining the two points.
Let $z_1(x_1,y_1),\:z_2(x_2,y_2),\:and\:z_3(x_3,y_3)$ be the vertices of
$A,B,C$ respectively of a parallelogran $ABCD$
Let $D=z_4(x_4,y_4)$ be the fourth vertex.
Diagonals of a parallelogram bisect each other (say at $E$)
Then $E$ is given by $(\large\frac{x_1+x_3}{2},\:\frac{y_1+y_3}{2})=(\frac{x_2+x_4}{2},\frac{y_2+y_4}{2})$
$i.e., \:E=\large\frac{z_1+z_3}{2}=\frac{z_2+z_4}{2}$
$\Rightarrow\:z_4=z_1-z_2+z_3$