Geometry question about lines

Question

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?

Answer 1

You could use the fact that two non-parallel lines intersect at one unique point, and so if the segment intersects the line, then one end point of the segment is "above" and the other is "below", otherwise you would have at least two intersections because a segment is continuous. Here "above" and "below" are taken in the weak sense, i.e. one of the points could be on the line.

Answer 2

It can be proved by contradiction.

Suppose that the target line going through the first point is not entirely on the same bank of the given line as those two points.

Case-1. (an impossible case) That line is entirely on the other bank. How can that line connect those points?

Case-2 Then it must have crossed the given line to reach the second point. Where could that second point be?

Answer 3

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.

Answer 4

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.