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Antecedent of Playfair's axiom: a line and a point not on the line Consequent of Playfair's axiom: a second line, parallel to the first, passing through the point. In geometry, Playfair's axiom is an axiom that can be used instead of the fifth postulate of Euclid (the parallel postulate):
For any point A and line l not incident with it (an anti-flag) there is exactly one line m incident with A (that is, A I m), that does not meet l (known as Playfair's axiom), and satisfying the non-degeneracy condition: There exists a triangle, i.e. three non-collinear points. The lines l and m in the statement of Playfair's axiom are said to ...
They are non-degenerate linear spaces satisfying Playfair's axiom. The familiar Euclidean plane is an affine plane. There are many finite and infinite affine planes. As well as affine planes over fields (and division rings), there are also many non-Desarguesian planes, not derived from coordinates in a division ring, satisfying these axioms.
If the sum of the interior angles α and β is less than 180°, the two straight lines, produced indefinitely, meet on that side. In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry.
Playfair's axiom. In a plane, there is at most one line that can be drawn parallel to another given one through an external point. (Proclus, 5th century, but popularized by John Playfair, late 18th century) The sum of the angles in every triangle is 180° (Gerolamo Saccheri, 1733; Adrien-Marie Legendre, early 19th century)
Playfair's axiom: Given a line and a point not on , there exists exactly one line ′ containing such that ′ =. There exists a set of four points, no three of which belong to the same line. The last axiom ensures that the geometry is not trivial (either empty or too simple to be of interest, such as a single line with an arbitrary number of ...
Therefore, Playfair's axiom (Given a line L and a point P not on L, there is exactly one line parallel to L that passes through P.) is fundamental in affine geometry. Comparisons of figures in affine geometry are made with affine transformations , which are mappings that preserve alignment of points and parallelism of lines.
Hilbert uses the Playfair axiom form, while Birkhoff, for instance, uses the axiom that says that, "There exists a pair of similar but not congruent triangles." In any of these systems, removal of the one axiom equivalent to the parallel postulate, in whatever form it takes, and leaving all the other axioms intact, produces absolute geometry.