Theorem 2.1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14

**Theorem 2.1**:
(The parametric representation of a line) Given two points
(*x*_{1}, *y*_{1}) and (*x*_{2}, *y*_{2}),
the point (*x*, *y*) is on the line determined by (*x*_{1},
*y*_{1}) and (*x*_{2}, *y*_{2}) if and only if
there is a real number *t* such that

and

**Theorem 2.2**:
(The parametric form of the Ruler Axiom) Let t be a real number. Let
*A* and *B* be two points. Let

and

Let *t* be a real number, and let

using vector addition and scalar multiplication of points. Then, the distance from *A* to *C,*

|*A**C*| = |*t*| |*A**B*|
,

where |*A**B*| is the distance from *A* to *B*, and the distance from *C* to *B*,

|*C**B*| = |1 - *t*| |*AB*| .

Which is to say that, if * C* is a point on the line segment between *A* and *B*, that

|*AB*| = |*AC*| + |*CB*|

**Theorem 2.3**:
If * C* is on the line segment between *A* and *B* then

If *C* is on the line determined by *A* and *B* but on the other side of *B* from *A* then

If *C* is on the line determined by *A* and *B* but on the other side of *A* from *B*, then

**Corollary:** (The midpoint
formula) Let (*x*_{1}, *y*_{1}) and (*x*_{2},
*y*_{2}) be two points. The midpoint between them has
coordinates^{1}

**Theorem 2.4**:
If * C* * * is on the line segment between * * *A* * * and * * *B* * * then * * *A* * * and * * *B* * * are on
opposite sides of * * *C*.

**Theorem 2.5**:
Let * * *A* * * be a point on the line determined by the equation * * *ax* + *by* = *c*,
* * and let * * *B* * * be a point not on that line. Then the points on the line
determined by * * *A* * * and * * *B* * * which are on the same side of * A* * * as * * *B* * * are on the
same side of the line * * *ax* + *by* = *c* * * as * * *B*, * * and the points on the other
side of * A* from * * *B* * * on the line determined by * * *A* * * and * * *B* * * are on the other
side of the line * ax* + *by* = *c*.

**Theorem 2.6**:
If two lines are parallel, then all of the points on one line lie on
the same side of the other line.

**Theorem 2.7**:
Given points * * *A* * * and * * *B ** * and a line whose equation is* ax + by = c,*

**Theorem 2.8**:
If a line segment contains points on both sides of another line, then
the line must intersect the segment somewhere between its endpoints.

**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

for the same value of * t*.

**Theorem
2.10**: Let * * *A*, * * *B*, * * and * * *C* * * be three noncolinear points. If * * *D* * * is on
the line through * * *A* * * which is parallel to * * *BC* * * then there is a real
number * * *s* * * such that

**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* * * such that

and

**Theorem
2.12**: Let * A*, * * *B*, * * and * * *C* * * be three noncolinear points, and let

be a point in the plane, where

Then * D* is on the same side of * * *BC* * * as * * *A* * * if
and only if * * *q* > 0.

**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 * * *AC*, * * then that line intersects the line segment * BC*.

**Theorem
2.14**: (The Second Pasch property) Let* A*, * * *B*, * * and * * *C* be three
noncolinear points. If a line intersects the line segment * AB*, * * then
the line will either intersect line segment * AC*, * * segment * * *BC*, * * or go
through point * * *C*.