Some Theorems of the Euclidean Geometry in Pentagonal Quasigroups
Pentagonal quasigroups are IM-quasigroups in which the additional identity of pentagonality holds. Motivated by the example C(q), where q is a solution of the equation \(q^4-3q^3+4q^2-2q+1=0\), some basic geometric concepts are defined in a general pentagonal quasigroup. Such concepts are parallelogram, midpoint of a segment, regular pentagon and regular decagon with their centres. The connection between pentagonal and, much better known, GS-quasigroups is mentioned. That connection enables introduction of more geometric concepts in pentagonal quasigroups. In this article some theorems of the Euclidean geometry which use all these concepts are stated and proved in pentagonal quasigroups.
KeywordsPentagonal quasigroup Parallelogram Regular pentagon Regular decagon