Prove directly from the axioms $(I1)-(I3)$ and $(B1)-(B4)$ that for any two distinct points A, B, there exists a point C with $A * C * B$.
My original thought was to use B2 to place a point D on the line AB s.t ABD holds then use it again to place a point E so that EBD holds but there is no guarantee doing this that E lies on the correct side of of A.
My next try was to use I3 to place a point F not on the line segment AB then connect AF and extend it with B2 to get G s.t $A*F*G$ now connect BG and we can use Paschs axiom on a line through the point F to say there is some point H either between BG or AB but this also become cyclic cause we can never guarantee that we place the point in between AB...
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 B are two points, then there exists at least one point C on the line AB such that B lies between A and C.
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.