I have this question:
$L_1$ and $L_2$ are lines, and $P$ is a plane. $L_1$ does not lie in $P$, but meets the plane in a point $p$. $L_2$ lies in $P$, but does not contain $p$. Does $L_1$ meet $L_2$? Which of the axioms of incidence are used in solving this question?
I think $L_1$ does not meet $L_2$ because $L_1$ and $L_2$ are not coplanar ?!
The axioms are:
- (line axiom) Through any two distinct point there is exactly one line,
- (plane axiom) Through any $3$-non collinear points there is exactly one plane
- (dimension axiom) Any line contains at least two distinct points, any plane contains at least two distinct lines, there are at least two distinct planes.
- (line-plane intersection) If two distinct points of a line lie in some plane $P$ then the whole line lies on P
- (the parallel axiom) Let $L$ be a line and $p$ be a point then there is exactly one line that passes through $p$ and is parallel to $L$.
- (plane plane intersection) If two distinct planes meet then their intersection is a line.