**Theorem 5.1**:
If two distinct points on
a circle are not
diametrically
opposed, then the
tangent
lines at those
points meet a
point
outside the
circle.

**Proof**: Let *A *and *B *be the two points on the
circle which are not
diametrically
opposed, and let *O *be the center of the circle.

If *OA *and *OB *determine the same line, since, by
Theorem 3.3, a
line and a
circle meet at at most two
points, *A *and *B *would either
have to be the same point or
be diametrically
opposed, which we are hypothesizing does not happen. So, if *OA *and *OB *determine distinct lines which share a
point, Then by
Theorem 1.6, they cannot
be parallel, so they have
different slopes. But the
line which is
tangent to the
circle at *A *is perpendicular to *OA *by Theorem 3.7.
Similarly, the line which is
tangent to the circle at *B *is perpendicular to *OB*. Since,
by definition, perpendicular lines have slopes which are
negative reciprocals of each other, the
tangent
lines have different
slopes, and so are not
parallel. Then by
Theorem 1.6 they meet.

Let's call the point where
they meet *C*. Since by Theorem
3.6, the shortest
distance from *O *to *AC *is
the perpendicular
distance which is the
distance from *O *to *A*, *C* is
farther away, and by definition will lie outside the
circle.

Analytic Foundations of Geometry