• Given any line, there are points on the line and points not on the line.

  • Given two distinct points there exists a unique line passing through those points.

  • Given three points on a line, one and only one of them is between the other two.

  • Given two points $A$ and $B$ there always exists a point $C$ between $A$ and $B$ and a point $D$ such that $B$ is between $A$ and $D$.

  • A line $m$ determines exactly two distinct semi planes, whose intersection is the line $m$.

Pasch's axiom: If a line not going through the vertices of a triangle (here I'm excluding the degenerate case of a triangle formed by three points on the same line) intersects one side, then it intersects another side.

  • 1
    $\begingroup$ Please define "semi-plane" $\endgroup$ – gebruiker Apr 13 '18 at 13:15
  • $\begingroup$ My foundations instincts are a little rusty, but the "semi-planes" postulate seems like a statement of the Plane Separation Axiom; the PSA is known to be equivalent to Pasch's Axiom, given an appropriate set of other axioms (say, Hilbert's). Off-hand, I couldn't say whether equivalence can be shown with the particular set of axioms you list. What's the context of this problem? A textbook exercise? $\endgroup$ – Blue Apr 13 '18 at 13:15
  • $\begingroup$ It is indeed a textbook exercise. Thanks for replying, I know now that the answer to my question is yes. $\endgroup$ – Matheus Andrade Apr 13 '18 at 13:23

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