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Introduction
Two circles can be constructed so that they either:
  • intersect at two points
  • intersect at one point (touch)
  • do not intersect

Suppose when there are two points of intersection that a line is drawn through these two points.

two circles

line through intersections
What will happen to this line when the two circles are moved apart?

Version 1 (see note below)
Move the points in this diagram so that the circles move apart. What happens to the line?

You will find that the two points of intersection and the line through them will disappear when the circles do not intersect.

Should this happen?

 Sorry, this page requires a Java-compatible web browser.

Version 2 (see note below)
Move the points in this diagram so that the circles move apart. What happens to the line?

You will find that the two points of intersection disappear but the line through them remains when the circles do not intersect.

Should this happen?

 Please enable Java for an interactive construction (with Cinderella).

A note

Version 1 above is produced by JavaSketchpad (the online version of Geometer's Sketchpad) and Version 2 by Cinderella. Look at what happens with CabriJava, C.a.R. and the Geometry applet and you will discover that Version 2, where the line does not disappear, occurs only with Cinderella.



How can the line still be shown when the points it passes through have both disappeared?
Is the line dependant on those points or on the circles for its existence?
Is the second display simply making a mistake?


The answers to these questions lie hidden in the mathematics behind the displays. This mathematics involves algebra, coordinate geometry, the solutions of quadratic equations, and the theory of complex numbers. The ideas lead on to concepts of the radical axis, geometrical inversions and more...

Follow these links to find out more...

>>> Intro | Quadratic equations | Coordinate geometry | Complex numbers | Conclusion

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