Theorem 2.9: Let A, B, and C be three noncolinear points, let D be a point on the line segment strictly between A and B, and let E be a point on the line segment strictly between A and C. Then DE is parallel to BC if and only if there is a nonzero real number t such that
and
Proof: The requirement that t be nonzero follows from the fact that both D and E are distinct from A.
Assume that there is such a real number t. Let
Then if we assume that there is a nonzero real number t such that
and
then
and
We must first dispose of the case where BC is vertical. In that case x1 = x2, but then, D and E can be seen to have the same x-coordinates as well, so DE will also be vertical, and thus be parallel to BC.
Otherwise, the slope of the line between D and E is
When we remove the parentheses in the top and bottom, the expression will simplify to
We can now factor
Since t is nonzero, we can cancel and get
which is the same as the slope of the line between B and C. As a result, by the definition of parallel lines, we conclude that the line between D and E is parallel to the line between B and C.
The converse follows from the uniqueness of the line through D parallel to BC, Theorem 1.7. We know that the line through D which contains the point E is parallel to BC. By Theorem 1.7. any line through D parallel to BC must then be the line determined by DE.