Theorems 4.1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12

**Theorem 4.1**:
(The distance axioms) Let *A*, *B*, and *C *be any three points in the
plane. Then

1. |*AB*| __>__ 0 with equality if and only if *A* = *B*.

2. |*AB*| = |*BA*|

3. |*AB*| __<__ |*AC*| + |*CB*| with equality if and only if *C* is on the line segment between *A *and *B*.

**Theorem 4.2**: If *M *is an isometry and *C *is a point on the line segment between *A *and *B*, then *M*(*C*) is a point on the line segment between *M*(*A*) and *M*(*B*).

**Theorem 4.3**:
Let *M *be an invertible isometry^{1}. If *A *and *B *are two points which are
on the same side of a line *L*, then *M*(*A*) and *M*(*B*) are on the same side
of *M*(*L*).^{2}

**Theorem 4.4**: If
we compose two isometries, the result is an isometry.

**Theorem 4.5**: A
translation is an isometry.

**Theorem 4.6**:
The reflection *R *of the point *A *in a line is

where *F *is the foot of the point *A *in the line.

**Theorem 4.7**:
The reflection *R *of the point (*x*_{1}, *y*_{1}) in the
line *y* = *mx* + *b* is

The reflection *R *of the point (*x*_{1}, *y*_{1}) in the
vertical line *x* = *a *is

**Theorem 4.8**:
A reflection is an isometry.

**Theorem 4.9**:
If *R* is a reflection, then *R*(*R*(*A*)) = *A *for all points *A *in the plane.

**Theorem
4.10**: If a point on a circle is reflected about a line that
goes through the center of the circle, then the image of the point is
also on the circle.

**Theorem
4.11**: If *T *is the translation

then the translation

is the inverse of
*T*.^{3}

**Theorem
4.12**: If two circles intersect, then the two points of
intersection are reflections of each other about the line joining
their centers.