Let $A,B$ be two points on opposite sides of a line $l$. Then the line segment $AB$ intersects $l$.
My question is:
Using only the 4 postulates of Euclid, is there a way to make precise the meaning of ``opposite sides"?
Is the intersection guaranteed to exist using only the 4 postulates? If so, why is it true?
Thanks!
The property you gave is the definition of opposite sides. We could say that A and B are on the same side if their line segment does not intersect l. This is an equivalence relation with only two classes. This follows from Hilbert's axiom system, but not using parallel postulate.