Theorem 2.11: (The parametric representation of a plane) Let A, B, and C be three noncolinear points. Let D be any point in the plane. Then there are real numbers q, r, and s with
Proof: Consider the line through D parallel to BC. There are two cases to consider. The first case is where this line contains A, and the other case is where it doesn't.
If the line through D which is parallel to BC contains the point A, then D is on the line through A which is parallel to BC, so, by Theorem 2.10, there exists a real number s such that
So, we can assume that the line through D which is parallel to BC does not contain the point A. Since AB and BC intersect at point B, by Theorem 1.6 they are not parallel, so any line which is parallel to BC is not parallel to AB, by Theorem 1.13, so there will be a point where the line through D, which is parallel to BC will meet the line determined by A and B by Theorem 1.6. Call this point E. Similarly, the line through D which is parallel to BC will meet AC at a point which we will call F.
In the illustration, D is being portrayed as being between E and F, but it does not have to be in the following proof.
If E and F were the same point, then since that point would be on the intersection of AB and AC, and by Theorem 1.6, the point of intersection is unique, that point would have to be A. Since we are considering the case where the line through D, parallel to BC does not contain A, this doesn't happen, so we can assume that E and F are different points.
Since EF is parallel to BC, by Theorem 2.9, there will exist a real number t such that
Since D is on the line determined by E and F, by Theorem 2.1, there will exist a real number u such that
If we substitute in the formulas for E and F we get
The A terms can be combined to
next theorem (2.12)