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