It is evident that the first construction is possible by compasses alone; just draw the two circles. We will first show that the other two constructions are possible by means of two other constructions, PC and CM, which we'll show later can be performed by compasses alone.
Step 1. Select an arbitrary circle outside the figure, that is, a circle containing no parts of the figure. Call it the bounding circle. Use PC to invert in that bounding circle all the specified points of the original figure. The inverted points will all lie inside the bounding circle. This inversion converts the data for circles and lines of the original figure to data for circles in the inverted figure.
Step 2. We'll need the centers of the circles in the inverted figure in order to draw the circles. The construction CN constructs those centers so long as we have three points on each circle, and we can get those byusing PC to invert enough points from the original circles and lines.
Step 3. Each required intersection of the original figure — whether of two circles, of a circle and a line, or of two lines — corresponds to an intersection of two circles in the inverted figure. Use CC to find the intersections in the inverted figure.
Step 4. Use PC again to invert the intersections in the inverted figure back to the intersections of the original figure.
Thus, the constructions CC, PC, and CN are sufficient to perform the constructions LC and LL.
The above diagram shows the original figure of two circles and two lines inverted in a bounding circle RST. The large circle RST on the right is outside the original figure, and the inverted figure is displayed inside it. Note how the blue and orange circles are inverted to blue and orange circles, but the lines GH and MN are inverted to (white) circles. All the straight lines from the original figure invert to circles passing through the center of RST. The inverted points A' and B' are not shown because they are not the centers of the inverted circles and play no role in the construction.
April, 1998; March, 2002.
David E. Joyce
Department of Mathematics and Computer Science
Clark University
Worcester, MA 01610
Email: djoyce@clarku.edu
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