I can't prove one theorem from Hilbert's "Foundations of Geometry". Here is the quote:
Theorem 6. Every simple polygon, whose vertices all lie in a plane $\alpha$, divides the points of this plane, not belonging to the broken line constituting the sides of the polygon, into two regions, an interior and an exterior, having the following properties: If $A$ is a point of the interior region (interior point) and $B$ a point of the exterior region (exterior point), then any broken line joining $A$ and $B$ must have at least one point in common with the polygon. If, on the other hand, $A$, $A'$ are two points of the interior and $B$, $B'$ two points of the exterior region, then there are always broken lines to be found joining $A$ with $A'$ and $B$ with $B'$ without having a point in common with the polygon. There exist straight lines in the plane $\alpha$ which lie entirely outside of the given polygon, but there are none which lie entirely within it.
This theorem can be proved with Hilbert's axioms of connection and order. No need to use congruence, parallel or continuity axioms.
Hilbert writes this theorem may be obtained without serious difficulty with the aid of this theorem:
Theorem 5. Every straight line $a$, which lies in a plane $\alpha$, divides the remaining points of this plane into two regions having the following properties: Every point $A$ of the one region determines with each point $B$ of the other region a segment $AB$ containing a point of the straight line $a$. On the other hand, any two points $A$, $A'$ of the same region determine a segment $AA'$ containing no point of $a$.
but for me it's not so obvious.
I had a few ideas to prove it but none of them was successful. I tried to indicate these regions as a sum of intersections of families of some half-planes and its complement. I also tried to show that a binary relation defined this way:
$A\sim B \iff$ there exists a broken line joining $A$ and $B$ which has no point in common with the polygon
is an equivalence relation with exacly two equivalence classes. I also tried to do this using induction. Can you help me?