Theorem 2.13: (The first Pasch property) Let A, B, and C be three noncolinear points. If a line going through A contains points in the angle between AB and BC, then the line intersects the line segment BC.
Proof: Suppose a line through A contains a point D which is inside the angle.
The illustration indicates that D can be on either side of the line determined by B and C, so long as it is inside the angle between AB and BC.
Then by Theorem 2.11, there are real numbers q, r, and s such that
where
Since by the definition of the angle, D is on the same side of AB as C, it follows from Theorem 2.12 that s is positive. Similarly, we can conclude that r is also positive. Since r and s are both positive we can define
Then by Theorem 2.1
is on the line determined by A and D. But
Find common denominators in the coefficient of the first term, and remove the parentheses in the second term.
Combine the A terms
But q + r + s = 1, so q + r + s - 1 = 0, and the A term has a zero coefficient. We get
However,
so E is on the line determined by B and C by Theorem 2.1. Moreover, since r and s are both positive s/(r + s) is between 0 and 1, so by the definition of the line segment between two points, E is on the line segment between B and C.