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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.

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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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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 ...

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  • BSc , Wuhan Univ. China
  • MA, Shandong Univ.
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  • "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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