Conclusion
To fully appreciate the mathematics involved in this exploration, you need to understand the sections on quadratic equations, coordinate geometry and complex numbers, and in particular the connections between intersections of lines, solutions of equations and complex conjugates.

Does the line through the points of intersection depend on those points for its existence, or is it essentially dependent on the two original circles only? We have seen already that the midpoint can always be displayed. So why not the line also? We have seen (Statement 2) that the line joining two points whose coordinates are complex conjugates is a real line.

Clearly, the line in question always passes through the midpoint and is always perpendicular to the line joining the centres of the circles. These two constructs always exist, regardless of whether the circles intersect or not, so therefore our line must always exist - though only the interactive geometry program Cinderella is equipped to display it.

To sum up...
Complex number theory and coordinate geometry agree. The line through the intersections always exists. This display shows the construction again, including the midpoint between the two intersections, and the line through the intersections. The window to the right shows the numerical values of the coordinates of the points and the equations of the circles and the intersection line.
Drag the circles so they are not intersecting, and look at the coordinates given for points E and F. You will see that they become complex values. The x coordinates are complex conjugates, as are the y coordinates. The midpoint G can always be plotted, even when E and F are complex.

Further work

1. The radical axis
This line through G which always exists even when the circles do not intersect is called the radical axis. It has the property that it is the set of all points with equal tangents to the two given circles. In the diagram below, tangents from P can be drawn to both circles. As long as P is on the radical axis all four tangents are of the same length.
(Also shown is the circle, centre P with radius equal to these tangents. It is orthogonal to the original circles. Find out what happens when those original circles are moved - in particular when one is inside the other or when they intersect. A related idea is that of inverse points.