Construction of a Regular Pentagon, Using Only a Compass

[sdi]

As a computer programmer, I'm enamored with the notion of economy: that is, doing as much as one can with as little as possible. It is perhaps because of this that I have always found geometry appealing: one endeavors to begin with the simplest tools possible—a collapsing compass and a straightedge—and, using these, build oneself ever-more-powerful tools, allowing one to easily draw figures of tremendous sophistication.

So it was with some amusement that I recently learned that one does not even need a straightedge: everything that can be constructed with a compass and straightedge can be constructed with a compass alone. (We're assuming that one can draw straight lines after the construction is complete to make the construction clear, but one cannot use a straightedge to find the endpoints of any of those lines!)

I haven't studied the proof of that in any depth, yet, but since we have a few fans of geometry here, I thought I might post one of my favorite compass-only constructions, which is due to Kurt Hofstetter.

Given. Points A and B.

I. Construct a circle from A to B.

II. Construct a circle from B to A.

III. Construct a circle between the two intersection points of circles I and II.

IV. Construct a circle from A to the intersection point of circles II and III not on circle I.

V. Construct a circle from B to the intersection point of circles I and III not on circle II.

VI. Construct a circle from one of the intersection points of circles IV and V to the center of circle III.

VII. Construct a circle from the other intersection point of circles IV and V to the center of circle III.

Result. The five points shown form a regular pentagon.