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202209291831 Problem 1.1

Prove the parallelogram law

|z_1+z_2|^2+|z_1-z_2|^2=2[|z_1|^2+|z_2|^2]

and give a geometric interpretation.

Extracted from R. B. Ash & W. P. Novinger. (2004). Complex Variables.

Draw a picture.

Apply the Law of Cosines,

\begin{aligned} |z_1+z_2|^2 & =|z_1|^2+|z_2|^2-2(-z_1)\cdot z_2 \\ |z_1-z_2|^2 & =|z_1|^2+|z_2|^2-2z_1\cdot z_2 \\ \end{aligned}

and that is all I could interpret.