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Points, straight lines, planes and their mutual position
The point, the straight line, the plane and the expression lying on are basic concepts, these are not defined. In Figure 1
- straight line \latex{ e } lies on plane \latex{S\left(e\subset S\right) };
- point \latex{ P } lies on straight line \latex{ e }, thus it also lies on plane \latex{S\left(P\in e,P\in S\right) };
- point \latex{ Q } lies on plane \latex{ S }, but it does not lie on straight line \latex{e\;\left(Q\in S,Q\notin e\right) };
- point \latex{ R } does not lie on plane \latex{S\left(R\notin S\right) }.
Figure 1
\latex{ Q }
\latex{ P }
\latex{ S }
\latex{ R }
\latex{ e }
Two distinct straight lines are
- intersecting, if they have a common point. (Figure 2)
- parallel, if they lie on one plane (are coplanar) and have no point in common. Besides that every straight line is parallel with itself. (Figure 3)
- skew, if they are not intersecting and not parallel/not coplanar. (Figure 4) E.g. the sides \latex{ AD } and \latex{ EF } of the cube shown in Figure 5 are skew lines.
Figure 2
\latex{e\cap f=M}
\latex{ M }
\latex{ e }
\latex{ f }
Two distinct planes are
- intersecting, if they have exactly one straight line in common. (Figure 6)
- parallel, if they have no point in common. Besides that every plane
is parallel with itself.
A straight line
- lies on a plane, if every point of the straight line is also a point of the plane.
- intersects a plane, if they have exactly one point in common.
- is parallel with a plane, if they have no point in common. (Figure 7)
Figure 3
\latex{e\cap f=\varnothing \left(e\parallel f\right) }
\latex{ e }
\latex{ f }
Based on the graphics we accept the following statements as true:
- Exactly one straight line passes through two distinct points.
- If three distinct points are not collinear, then there is exactly one plane which contains the three points.
- If two distinct points of a straight line lie on a plane, then every point of the straight line lies on the plane.
- There is exactly one plane containing a given straight line and a point not lying on the straight line.
- Exactly one straight line passes through a given point so that it is parallel with a given straight line.
Figure 4
\latex{ e }
\latex{ f }
Figure 5
\latex{ F }
\latex{ B }
\latex{ A }
\latex{ C }
\latex{ G }
\latex{ E }
\latex{ H }
\latex{ D }
Figure 6
\latex{S_{2}}
\latex{S_{1} \cap S_{2}=e }
\latex{S_{1}}
\latex{ e }
Figure 7
\latex{e\subset S}
\latex{f\cap S=M}
\latex{g\cap S=\varnothing }
\latex{\left(g\parallel S\right) }
\latex{ e }
\latex{ S }
\latex{ M }
\latex{ f }
\latex{ g }