**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

such that

**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

where

so

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.

Let

and

Since * * *EF* * * is parallel
to * * *BC*, * * by Theorem 2.9, there will
exist a real number * * *t* * * such that

and

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

Distribute

The * * *A* * * terms can be combined to

Let

and

Then

and