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The Napoleon-Barlotti theorem in hexagonal quasigroups

Hexagonal quasigroups are idempotent medial quasigroups in which the additional identity of semisymmetricity, a b ⋅ a = b ab⋅a=b, holds. The famous Napoleon-Barlotti theorem of Euclidean geometry says: The centres of the regular n n- gons constructed on the sides of an affine regular n n-gon form a regular n n-gon. In this talk the concepts of parallelogram, regular triangle and its centre, regular hexagon and its centre and affine regular hexagon are introduced in hexagonal quasigroups. Some illustrations of these concepts are given in the model C ( q ) C(q), where q q is a solution of the equation q 2 − q + 1 = 0 q2−q+1=0. The Napoleon-Barlotti theorem in the cases n = 3 n=3 and n = 6 n=6 is stated and proved in a general hexagonal quasigroup.

IM-quasigroup, hexagonal quasigroup, regular triangle, regular hexagon

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