I'm studying axiomatic Euclidean Geometry, and I'm using two books which have slightly different set of axioms. Here's the set of Incidence Axioms and Betweenness Axioms of the first one:

I1) For any line, there are at least two points in the line and two points outside of it.

I2) For any given two distinct points, there is a unique line that contains both of them.

B1) For any given three distinct points in a line, one and only one is between the other two.

B2) For any two distinct points, there are points $C$ and $D$ such that $C$ is between $A$ and $B$, and $B$ is between $A$ and $D$.

B3) A line determines exactly two semiplanes.

Where, by definition, given a line $l$ and a point $A$, the semiplane determined by $l$ and $A$ is the set of points $B$ such that $AB$ (the set of points between $A$ and $B$) does not intersect the line l.

Using that, I'd like to prove Pasch's Axiom:

"Let $A$, $B$, $C$ be three points that do not lie on a line and let $l$ be a line which does not meet any of the points $A$, $B$, $C$. If the line $l$ passes through a point of the segment $AB$, it also passes through a point of the segment $AC$, or through a point of segment $BC$."

Is it possible? I'm not sure if it's provable without using the later axioms of the book.


Consider the two half-planes determined by line $l$. Points $A$ and $B$ belong to different half-planes, because $AB$ intersects $l$. It follows that either $C$ belongs to $l$, or it belongs to the same half plane as $A$ (and then $BC$ intersects $l$), or it belongs to the same half plane as $B$ (and then $AC$ intersects $l$).


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