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?

Given an equation for the line $f(x) = ax+b$, you can define a function $g:[c,d]\rightarrow\mathbb{R}$ that connects the two lines. Write the distance between the lines as a function of $x$ and then check for local extrema, you will find that there are non, therefore the minimal distance has to be at the global points $c,d$ which are just the original points. Thus every point on the line segment has to be on the same side.
