Theorem 2.14: (The second Pasch property) Let A, B, and C be three noncolinear points. If a line crosses the line segment AB, then the line will either intersect line segment AC, segment BC, or go through point C.
Proof: Let D be the point where the line crosses line segment AB. We can conclude from Theorem 2.1 that there are points on the line segment between A and B, which are on either side of the point D. By Theorem 2.5, then, there are points on that line segment which are on either sides of the line determined by A and B. Let E be a point on the same side of the line determined by AB as C.
Consider the line determined by CD. Since C is, by hypotheses, not on the line determined by AB, we conclude that CD crosses AB at D. As a result, by Theorem 2.5, A is on one side of the line determined by CD, and B is on the other side.
If we substitute the coordinates of E into the equation for the line determined by CD, the trichotomy law for real numbers states that there are three cases.
Case 1: E is on CD. Then the line determined by CD and the line determined by DE both contain the distinct points D and E, so by Theorem 1.4, they are the same line, and the line determined by DE contains the point C.
Case 2: E is on the same side of CD as A. We also have that E is on the same side of AD as C, because E is supposed to be on the same side of AB as C, and AD has the same equation as AB, so E is on the same side of AD as C. The hypotheses of Theorem 2.13 are satisfied, so we can conclude that DE crosses the line segment between A and C.
Case 3: E is on the same side of CD as B. This case is the same as Case 2 if we simply switch A's and B's, and the conclusion is that DE crosses the line segment between B and C.