The identity is called the parallelogram law because if we think of z and w as vectors (or points) in the plane, then it tells us that the sum of the squares of the lengths of the sides of a parallelogram is equal to the sum of the squares of the lengths of its diagonals.
To prove the identity, just write
The last equality follows from .
Similarly, we have
Adding these gives the identity.
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Author: Michael O'Neill