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The axiom in this section caused the most controversy and confusion of
all. The axioms of parallels (which is also an incidence axiom) is
**Axiom of Parallels**
- Given a line and a point outside it there is
exactly one line through the given point which lies in the plane of the
given line and point so that the two lines do not meet.

Note that, while asserting that there *is* a line through the
given point that doesn't meet the given line, it also says there is
only one such line. In other words, it also asserts that all the
``other'' lines co-planar with the given line meet that line. This
motivates the introduction of the following (stronger and stranger)
version of the Axiom of Parallels:
**Projective Axiom of Parallels**
- Any pair of lines that lie in the
same plane meet.

The idea behind this axiom is that even (apparently) parallel lines
appear to meet at the horizon. We can demonstrate that this axiom is
consistent with the axioms of Incidence by means of
Linear Algebra as in the examples below.

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Kapil H. Paranjape
2001-01-20