For two vectors prove that we have the identity

What geometric theorem can you deduce from this identity?

*Proof.* We compute

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Stumbling Robot

A Fraction of a Dot
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Prove an identity for two vectors in Euclidean *n*-space

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For two vectors prove that we have the identity

What geometric theorem can you deduce from this identity?

*Proof.* We compute

Is the geometric identity the one that the sum of squares of diagonals equals two times the sum of squares of non paralel sides ? To be more precise this one :- https://en.wikipedia.org/wiki/Parallelogram_law

Yes, as written at the back of the book (i.e. there are 4 sides in total).