Geometry of Pentagonal Quasigroups

Stipe Vidak

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 introduced and studied in a general pentagonal quasigroup. Such concepts are parallelogram, midpoint of a segment, regular pentagon and regular decagon. Some theorems of Euclidean plane which use these concepts are stated and proved in pentagonal quasigroups.