Greek constructions and GSP constructions

Ancient Greek geometers in the days of Euclid (about 300 BC) thought a lot about the problem of constructing geometric figures in the plane. Their procedure is usually called construction with ruler and compass, but a more exact terminology would be construction with straight-edge and compass, since they didn't allow using the markings on a ruler. (Greek constructions didn't involve measurement.)

The Greek rules of construction are as follows:

Start with two points A and B in the plane. We want to determine which points, lines, and circles can be "constructed" (from A and B).

1. If two points C and D can be constructed, then the line through C and D can be constructed. (This is the ruler or straight-edge rule.)

2. If two points C and D can be constructed, then the circle with center C which passes through D can be constructed. (This is the compass rule.)

3a. If two lines m and n can be constructed, and they have a point in common, then the intersection point of m and n can be constructed. (This is the line intersection rule.)

3b. If two circles c and d can be constructed, and they have one or two points in common, then the points of intersection of c and d can be constructed. (This is the circle intersection rule.)

3c. If a line m and a circle c can be constructed, and they have one or two points in common, then the points of intersection of m and c can be constructed. (This is the line-circle intersection rule.)

For example: To construct a regular pentagon means to start with two points A and B and to construct three more points X, Y, Z, so that the polygon ABXYZ is a regular pentagon.

The Greek construction rules correspond to certain commands in the Geometer's Sketchpad Construct menu:
 

• Rule 1 corresponds to the GSP command Line.
• Rule 2 corresponds to the GSP command Circle By Center+Point.
• Rule 3 corresponds to the GSP command Intersection.

Most of the other commands in the Construct menu can be done by using only the three basic commands Line, Circle By Center+Point, Intersection. The commands which can be done using these three basic commands are Midpoint, Parallel Line, Perpendicular Line, Angle Bisector, Circle By Center+Radius, Arc Through 3 Points. (More precisely, "Circle Through 3 Points" can be done with the three basic commands. The relation between an arc and a circle is analogous to the relation between a segment and a line.)

The commands Point on Object and Locus in the GSP Construct menu cannot be done using the Greek rules. The command Point on Object chooses a random point on a line, ray, segment, circle, etc. The Greek rules don't allow such a choice. The Locus command uses the Point on Object command. In fact, the Locus command can be used to construct curves other than circles (like ellipses), and such curves are outside the realm of Greek constructions. Locus is a dynamic construction tool, in constrast to the static construction tools of the Greeks.

Note: Some GSP commands are equivalent or inessential from the Greek viewpoint. The commands Segment and Ray are considered to be equivalent to the command Line. The command Arc on Circle is considered to be equivalent to Arc Through 3 Points. The command Interior is inessential; i.e. the interior of a figure was considered by the Greeks to be constructed as soon as the figure is constructed. The reason GSP has commands like Segment and Interior is so that the Measure menu can be used to compute lengths and areas.