Let [a, b; c, d] stand for the statement ‘abdc forms a parallelogram’ (where a, b, d, and c are taken cyclicly). Take as axioms (i) for any a, b, and c, there exists d such that [a, b; c, d ]; (ii) if [a, b; c, d ], then [b, a; d, c] and [a, c; b, d ]; (iii) if [a, b; c, d ] and [a, b; e, f ], then [c, d; e, f ]. Show that, when any chosen point is singled out and labelled as the origin, this algebraic structure reduces to that of a ‘vector space’, but without the ‘scalar multiplication’ operation, that is to say, we get the rules of an additive Abelian group.
Can anyone please point out how can I approach this question? Any help will be appreciated.



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