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

**Proof**: By Theorem
3.13, if two circles
intersect, then there are either one or two
points of intersection. If
there is one point of
intersection, then by Theorem
3.12, it is on the line
joining the two centers, so
is fixed by a reflection about that
line. If there are two
points of intersection,
then the line determined
by the two points of
intersection is
perpendicular to
the line joining their
centers by
Theorem 3.11. Therefore,
the line joining the
centers intersects the
line segment
between the points of
intersection at its
midpoint by
Theorem 3.9. So to get from
one of the points of
intersection to the other, you travel from one
point of intersection
perpendicular to
the line joining their
centers and then continue an
equal distance beyond to get to the next point. This is the
definition of a reflection.

5. Arc Length, Angles, and Rotations