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# Points in a Complex Plane that are the Vertices of a Parallelogram

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Find necessary and sufficient conditions (with proofs) such that the points z1, z2, z3, and z4 in the complex plane are the vertices of a parallelogram. I have read that the points z1, z2, z3, and z4 in the complex plane are vertices of a parallelogram if and only if z1 + z3 = z2 + z4. But, if this is indeed the case I would like to see the proofs of both of the directions. Please explain your reasoning and solution in as much detail as possible.

https://brainmass.com/math/complex-analysis/complex-plane-vertices-parallelogram-12016

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Find necessary and sufficient conditions (with proofs) such that the points , , and in the complex plane are the vertices of a parallelogram.

Assume that four complex numbers , , and correspond to four points A,B,C and D, respectively. See the graph above. Let , then we have

Claim. ABCD is a parallelogram if and only if
Proof: We need to prove "If ABCD is a parallelogram, then ". Since ABCD is a parallelogram, we know that the ...

#### Solution Summary

The necessary and sufficient conditions (with proofs) such that the points z1, z2, z3, and z4 in the complex plane are the vertices of a parallelogram are shown. The solution is detailed and well presented.

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