**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 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 * * *r*/(*r* + *s*) * * and * s*/(*r* + *s*) * * are both
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*.