### Definitions

**(Vector addition of points)**
Let * * *A* = (*x*_{1}, *y*_{1}) * * and let * * *B* = (*x*_{2},
*y*_{2}) * * be two points.
Define

*A* + *B* = (*x*_{1} + *x*_{2}, *y*_{1} +
*y*_{2})
**(Scalar multiplication of points)**
Let * A* = (*x*, *y*) * * and let * * *r* * * be a real number. Define

*rA* = (*rx*, *ry*)
Let * * *A* * * and * * *B* * * be two points. The **line segment**
between * A* * * and * * *B* * * is the set of points of the form

*C* = (1 - *t*)*A* + *tB*
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where

0 __<__ *t* __<__ 1
*A* * * and * * *B* * * are called the **endpoints** of
the line segment between * * *A* * * and * * *B*.

The set of
points of the form

*C* = (1 - *t*)*A* + *tB*
where* t* > 1 * * are the points on **the other
side** of * B* * * from * * *A*, * * and the set of such points where * * *t* < 0
* * are the points on **the
other side** of * * *A* * * from * * *B*.

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The set of
points of the form

(1 - *t*)*A* +* tB*
where * * *t* > 0 * * is the set of points on the line determined by * * *A* * * and * B* * * that are on the **same side** of * * *A* * * as * * *B*.

The set of
points of the form

(1 - *t*)*A* + *tB*
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where * * *t* < 0 * * is the set of points on the
line determined by * * *A* * * and * * *B* * * which are on the **opposite side** of * * *A* * *than * * *B*.

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The set of points
of the form

(1 - *t*)*A* + *tB*
where * * *t* > 0 * * is called the **ray** from * * *A* * * through * * *B*.

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Given a
line whose equation is

*ax* + *by* = *c*
the set of points whose
coordinates satisfy

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*ax* + *by* < *c*
is the set of points
**one side of the line**, and the set of
points whose coordinates
satisfy

*ax* + *by* > *c*
is the set of points on
the **other side of the line**.

Let * * *A*, * * *B*, * * and * * *C* * * be three noncolinear points. The **angle**
between * * *AB* * * and * * *AC* * * is the set of points which are on the
same side of * AB* * * as* C *and the same side of * * *AC* * * as * * *B*. * * If a point is in the angle between
* * *AB* * * and * * *AC*, * * it is said to be **inside of the angle**.
Points which are not inside
of the angle, and not on the rays starting at * A* * * and going through * * *B* * * and * * *C*, * * are said to be **outside of the angle.**

A **triangle** is a figure determined by
three points consisting of
the three line
segments joining the three points. The three points are called the three **vertices**, and the line segments between
the vertices are called the **sides** of the triangle.

Let * * *A*, * * *B*, * * and * * C * * be three noncolinear points. The **interior of
the *** * triangle *ABC* * * is the set of points which are on the
same side of * AB* * * as * * *C*, the same side of * * *AC* * * as * * *B*, and the same side of * * *BC* * * as * * *A*. Points in the interior of the
triangle are said to be **inside of the triangle**.
Points which are not inside
the triangle are said to be **outside of the triangle**.

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