Editing Parallel (geometry) (section)

==Reflexive variant==
If ''l, m, n'' are three distinct lines, then <math>l \parallel m \ \land \  m \parallel n \ \implies \ l \parallel n .</math>

In this case, parallelism is a [[transitive relation]]. However, in case ''l'' = ''n'', the superimposed lines are ''not'' considered parallel in Euclidean geometry. The [[binary relation]] between parallel lines is evidently a [[symmetric relation]]. According to Euclid's tenets, parallelism is ''not'' a [[reflexive relation]] and thus ''fails'' to be an [[equivalence relation]]. Nevertheless, in [[affine geometry]] a [[pencil of parallel lines]] is taken as an [[equivalence class]] in the set of lines where parallelism is an equivalence relation.<ref>[[H. S. M. Coxeter]] (1961) ''Introduction to Geometry'', p 192, [[John Wiley & Sons]]</ref><ref>[[Wanda Szmielew]] (1983) ''From Affine to Euclidean Geometry'', p 17, [[D. Reidel]] {{isbn|90-277-1243-3}}</ref><ref>Andy Liu (2011) "Is parallelism an equivalence relation?", [[The College Mathematics Journal]] 42(5):372</ref>

To this end, [[Emil Artin]] (1957) adopted a definition of parallelism where two lines are parallel if they have all or none of their points in common.<ref>[[Emil Artin]] (1957) [https://archive.org/details/geometricalgebra033556mbp/page/n63/mode/2up?view=theater ''Geometric Algebra'', page 52] via [[Internet Archive]]</ref>
Then a line ''is'' parallel to itself so that the reflexive and transitive properties belong to this type of parallelism, creating an equivalence relation on the set of lines. In the study of [[incidence geometry]], this variant of parallelism is used in the [[affine plane (incidence geometry)|affine plane]].