Theorems 3.1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16

**Theorem
3.1**: The equation of the circle whose center is
(*x*_{0}, *y*_{0}) and whose radius is *r *is

**Theorem
3.2**: The circle whose equation is

and the vertical line *x* = *a *meet at the points where

**Theorem
3.3**: The circle whose equation is

and the line

meet at the points where

and

or

and

**Theorem 3.4**:
(Pythagoras)^{1} Let *f*, *d*, and *r *be the three sides of a right triangle, *r *the hypoteneuse.

Then

**Theorem 3.5**:
(The Triangle Inequality) Let *A*, *B*, and *C *be three points in the
plane. Then the distance from *A *to *B *is less than or equal to the sum
of the distances from *A *to *C *and *B *to *C *with equality if and only if *C *is on the line segment between *A *and *B*.

**Theorem 3.6**:
The shortest distance from a point to a line is the perpendicular
distance.

**Theorem
3.7**: A tangent line to a circle is perpendicular to the
radius to the point of tangency.

**Theorem 3.8**: If
a line is tangent to a circle, then all of the points which are
either on the circle or inside the circle except for the point of
tangency are all on the same side of the line.

**Theorem 3.9**:
Let *A *and *B *be two points on a circle. The foot of the center of the
circle in the line determined by *A *and *B *is the midpoint of the line
segment between *A *and *B*.

**Theorem
3.10**: The points which lie on the intersection of two circles
which have different centers whose equations are

and

lie on the line whose equation is

if the line joining the centers is horizontal and

otherwise

**Theorem
3.11**: The line joining the two points where two circles
intersect is perpendicular to the line joining their centers.

**Theorem
3.12**: The point where the line joining the centers of two
circles meets the line containing the points of intersection of the
two circles is given by

and

where (*x*_{1}, *y*_{1}) and (*x*_{2},
*y*_{2}) are the centers of the circles, *r*_{1 }and *r*_{2 }are the respective radii, and *d* is the distance between the centers.^{2}

**Theorem
3.13**: Given two circles whose equations are

and

their points of intersection are given
by^{3}

and

where

**Theorem 3.14**: If two circles have the same radius, then the points of intersection
between the two circles lie on the perpendicular bisector of the line
segment joining the two centers.^{4}

**Theorem 3.15**: Given a circle with center (*x*_{0}, *y*_{0}) and radius *r *and a point (*x*_{1}, *y*_{1}),

**Theorem 3.16**: Given a point * A* * * not on a line, any point whose distance from * A* * * is
less than the distance from * * *A* * * to its foot in the line is on the same side of the line as * A*.

4. Translations and Reflections