geometry – Law of parallelogram Geometric proof

I've therefore approached the parallelogram law in many ways over and over again now. And algebraically (or maybe I mean arithmetically) it seems perfectly logical to me – I can prove it, understand it and accept it. However, most authors use a representation of a parallelogram with diagonals or drawn vectors, like that of Wikipedia:

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This makes sense to me, but in the end, the parallelogram law connects the squares of these segments, not the segments themselves (directly), so I have no intuition to draw from this diagram. Can any one extend a geometric light on this subject without resorting to algebra (or with a minimal call)? If this is the standard proof (that is, similar to the Wikipedia proof), that does not interest me. Alternatively, if someone can show why my request is not possible, it would cool too.

Thank you.