Department of Mathematics
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Starting from the classical example of the "parallelogram law of addition" in the euclidean plane, we formally develop certain first order properties which define the algebraic concept of a reflection quasigroup. From these axioms we prove a quasigroup analog of the well-known geometric theorem that the midpoints of any quadrilateral form a parallelogram. The terminology is geometric in nature, we even draw pictures but the proofs are strictly algebraic.

1. F. Bachmann and J. Boczeck, Points, Vectors and Reflections. Foundations of Mathematics, Volume II, Geometry, M.I.T. Cambridge 1983.