Geometry question about lines If I have two points in euclidean space or the Cartesian plane whichever and both points lie on the same side of a straight line. Both above or both below- how can I show that the segment connecting the two points also lies above or below the line respectively . I.e every point on the segment is above or below the line respectively. This is so obviously true. Is one supposed to take it as axiomatically true? Or can it be proved?

The original line partitions the plane into $2$ subspaces. Given $2$ points, consider the line passing through them both. The original line partitions this line, and so if the points lie in the same partition, they are on the same side of the original line, and clearly the segment between them doesn't cross the original line.
