Glasnik Matematicki, Vol. 51, No. 2 (2016), 359-377.

THE NAPOLEON-BARLOTTI THEOREM IN PENTAGONAL QUASIGROUPS

Stipe Vidak

Department of Mathematics, University of Zagreb, 10 000 Zagreb, Croatia
e-mail: svidak@math.hr

Abstract.   Pentagonal quasigroups are IM-quasigroups in which the additional identity (ab · a) b · a=b holds. GS-quasigroups are IM-quasigroups in which the identity a(ab · c) · c=b holds. The relation between these two subclasses of IM-quasigroups is studied. The geometric concepts of GS-trapezoid and affine regular pentagon, previously defined and studied in GS-quasigroups, are now defined in a general pentagonal quasigroup. Along with the concepts of the regular pentagon and the centre of the regular pentagon, previously defined in pentagonal quasigroups, this enables formulations and proofs of some theorems of the Euclidean plane in a general pentagonal quasigroup. Among these theorems is the famous Napoleon-Barlotti theorem in the case n=5.

2010 Mathematics Subject Classification.   20N05.

Key words and phrases.   IM-quasigroup, pentagonal quasigroup, regular pentagon, affine regular pentagon.


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DOI: 10.3336/gm.51.2.06


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