**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.

**Proof**: It will suffice to show that all of the
points which are on the
other side of
the line from the
center of the
circle are
outside the
circle.

Let *O *be the center of the circle, let *P *be the point of
tangency between the
line and the
circle, and let *A *be any point on the
other side of
the line from *O*. Since *O *is
on one side of
the tangent
line and *A *is on the other side,
there is some point where
the line between *A *and *O *meets the tangent
line, by
Theorem 2.8. Call
this point* B*.

By Theorem 2.8, *B *is between *O *and *A*. So

by Theorem 2.3,
but by Theorem 3.7, the
line which is
tangent to the
circle at *A *is perpendicular to
the radius from *O *to *A*, and, by Theorem 3.6,

with equality if and only if *P* = *B*. Since |*BA*| is nonnegative we
have