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## the Axioms of Incidence

list the ways in which the elements interact (i.e., are "incident" upon, or with, each other), and these axioms control constructions.
Some follow.

1. Two points must have a line in common
2. Two planes must have a line in common
3. Two lines can, but need not, have a plane in common
4. If two lines have a plane in common, they must have exactly one point in common—never more, and never none
5. If two lines do not have a plane in common, they are skew, and have nothing at all in common
6. Three points have exactly one plane in common—never more, and never none
7. Three planes have exactly one point in common—never more, and never none

Note the general (and remarkable) interchangeability here
of point and plane, or of point and line.

Compare, for example, axioms (6) and (7).
This is called Duality.

We emphasise that, by (4) in this list, lines with a common plane must meet , and that, by (5),
lines that do not meet cannot be other than skew.

Animation illustrating that Only Skew Lines Do Not Meet
Please enable Java for an interactive construction (with Cinderella).

We emphasise, too, that taking an element common to other elements to be part of
(in the sense of “incorporated into”) these other elements
is a fundamental error.

For example,

• the point common to two lines is simply the place where the lines meet; the meeting-place is part of neither line,
because lines are not points.
• the line common to two planes is simply the line in which the planes meet; the meeting-line is part of neither plane,
because planes are not lines.

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