The PSA states that: A line $l$ separates a plane into two separate disjoint regions such that:
i) If points $P$ and $Q$ are in the same region, then the segment $\overline{PQ}$ does is fully contained in this same region.
ii) If points $P$ and $Q$ are in different regions, then the segment $\overline{PQ}$ intersects $l$.
Now, in Taxicab Geometry, it is easy to see that $i)$ is not true since you can have a line $l$ and two points $P$, $Q$ on the same side of $l$ and draw another line that joins $P$ to $Q$ passing through a point $R$ in the other side of $l$ (a quick drawing is enough to see this). This, of course, is keeping in mind that in taxicab geometry a line is sort of like a staircase.
I'm interested in finding a way to prove that ii) is true all the time. This time I don't just want to do a drawing because that would be for just one case, whereas we want to prove it true for all cases.
Any help is much appreciated.
The usual modern approach in geometry is to treat point and line as undefined terms and develop a geometry from axioms. From this point of view taxicab geometry has as one of its axioms that two points determine a unique line. The usual model for taxicab geometry is that one interprets points as ordered pairs of real numbers and lines as linear equations, so the points and lines in the taxicab plane and the euclidean plane are the same from this point of view. What differs is the way distance between points works, so in the taxicab plane the distance from (a,b) to (c,d) is given by |a-c| + |b-d|. Typically, unlike for the euclidean plane, there are many ways to get from (a,b) to (c,d) that have the same length - but there is only one line between the points. Note, however, what is true for the distance from (0,0) to (3,0), say. Only, the line determined by these points achieves the length 3, unlike what is true for (0,0) to (3,3), where many "routes" have length 6. The axiom that the two geometries differ in is the congruence axiom, which if taken to be SAS does not hold for the model described above. As a result many theorems in euclidean geometry are no longer theorems in texicab geometry.