Question
The points $${z_1},{z_2},{z_3},{z_4}$$ in the complex plane are the vertices of a parallelogram taken in order if and only if
A.
$${z_1} + {z_4} = {z_2} + {z_3}$$
B.
$${z_1} + {z_3} = {z_2} + {z_4}$$
C.
$${z_1} + {z_2} = {z_3} + {z_4}$$
D.
None of these
Answer :
$${z_1} + {z_3} = {z_2} + {z_4}$$
Solution :
If vertices of a parallelogram are $${z_1},{z_2},{z_3},{z_4}$$ then as diagonals bisect each other
$$\eqalign{
& \therefore \,\,\frac{{{z_1} + {z_3}}}{2} = \frac{{{z_2} + {z_4}}}{2} \cr
& \Rightarrow \,\,{z_1} + {z_3} = {z_2} + {z_4} \cr} $$