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  4. The points z1, z2, z3, z4 in the complex plane are the vertices of a parallelogram taken in order, iff :

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Q. The points $z_{1},\, z_{2},\, z_{3},\, z_{4}$ in the complex plane are the vertices of a parallelogram taken in order, iff :

Bihar CECEBihar CECE 2003

Solution:

Diagonals of a parallelogram bisect each other at the same point.
Since, the points $z_{1}, z_{2}, z_{3}$ and $z_{4}$ are the vertices of a parallelogram, therefore diagonals of a parallelogram bisect each other at a point
$\frac{z_{1}+z_{3}}{2}=\frac{z_{2}+z_{4}}{2}$
$\Rightarrow z_{1}+ z_{3}=z_{2}+z_{4}$