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?
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$\begingroup$ Maybe you could use the fact that "the line segment between two points is the shortest path"? $\endgroup$– 3x89g2Jul 22, 2015 at 15:15
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2$\begingroup$ Depending upon the set of axioms you're using, you may have an explicit "Plane Separation Postulate" that says exactly what you want. For instance, the School Mathematics Study Group axioms have it. But some systems use Pasch's Axiom, which (again, depending upon the other axioms in play) is logically equivalent to the Plane Separation Postulate. At the Foundations level, you have to be really careful about your assumptions. $\endgroup$– BlueJul 22, 2015 at 15:30
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$\begingroup$ @Blue Yeah I'm with you, it does depend on the assumptions and the OP is very close to basic assumptions. I used the assumption that every two non-parallel lines intersect at one point, but then I ended up needing continuity of a segment to complete the argument, which almost surely requires many more of the "simplest" assumptions. $\endgroup$– user2566092Jul 22, 2015 at 15:38
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4 Answers
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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?
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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.
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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.
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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.
