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Triangles on the Same Base (Or Equal Bases) and Equal Areas Will Lie between the Same Parallels

The complex n...

Question

The complex numbers $z_{1}, z_{2}, z_{3}$ are the vertices of a triangle. Find all the complex numbers $z$ which make the triangle into a parallelogram.

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Solution

If they represent a parallelogram then midpoint of the diagonal coincides
$⟹ \frac{(z_{1} + z_{3})}{2} = \frac{(z_{2} + z_{4})}{2}$
$⟹ (z_{1} + z_{3}) = (z_{2} + z_{4})$

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