# Equivalence of certain geometry axioms.

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.

## 1 Answer

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 $l$).