using Hilberts axioms Incidence, Betweenness and the line (plane) separation property show that the four points A,B,C,D behave as we expect with respect to between.
Proposition 6.2 simply says that two line intersect at at most one point and follows from the above axioms.
$A*B*C$ says B is between A and C.
Show that $A*B*C$ and $B*C*D$ imply $A*B*D$ and $A*C*D$
By B1 $A*B*C$ A,B,C are on a line by B1 $A*B*D$ are A,B,D are on a line by I1 they are on the same line since A,B are on both on it.
By B3 only one of the following can hold. $A*B*D$ or $A*D*B$ or $B*A*D$
Pick F not on the line and draw the line AF by line separation and $A*B*D$ we have BD is a line segment on one side or the other of AF as I1 says AF and BD are unique lines by choice of F they intersect only at A by prop 6.2 this implys that B~D by def'n however $B*A*D$ says B does not wiggle D hence it not a possible choice.
Now draw the line DF we can see that from $B*C*D$ that line segment BC is on one side or the other of DF as BC and DF are unique lines by I1 that intersect at D and only D by prop 6.2 hence B~C and from $A*B*C$ using the fact that D is the only point of the line AB and on DF we have that AC is connected by a line segment on one side or the other of D as B is between A and C so they must be on the same side of the line DF. But $A*D*B$ implys that A cannot wiggle B as they are on opposite side of the line DF. Hence the only possibility by B3 is $A*B*D$
am i missing some easy way of doing this? does this work?
I1)For every two points A and B there exists a line a that contains them both.
I2)There exist at least two points on a line.
I3)There exist at least three points that do not lie on the same line.
Axioms of betweeness
B1) If a point B lies between points A and C, B is also between C and A, and there exists a line containing the distinct points A, B, C.
B2) If A and C are two points, then there exists at least one point B on the line AC such that C lies between A and B.
B3) Of any three points situated on a line, there is no more than one which lies between the other two.
B4) Pasch's Axiom: Let A, B, C be three points not lying in the same line and let a be a line lying in the plane ABC and not passing through any of the points A, B, C. Then, if the line a passes through a point of the segment AB, it will also pass through either a point of the segment BC or a point of the segment AC.