The Official Rules

The rules below are actually a little more flexible than the ``official'' rules, but they are easy to understand, and the difference between these rules and the official rules is explained in the footnote. Here they are:

A construction problem begins with a set of given points, lines, and circles, and with some desired point, line or circle to construct, based on the given objects. In the example in Section 1, we were given two points and the segment between them, and the object of the construction was to find the point that lies on the segment midway between the two original points.

Notice that we could reduce the statement above to require that only certain points be constructed. Obviously, if you want a line, you can simply require that two different points be found on it, or if you want a circle you can require that the center and a point on the circle be found.

At any stage in the construction, you may do any of the following things to obtain additional points, lines or circles:

1. You may draw a straight line of any length through two existing points. (This means, of course, that the straightedge is as long as you need it to be, so it is better than a real ruler in that sense.)

2. You may find a new point at the intersection of two lines, two circles, or of a line and a circle. When you are given a segment, of course, you are given the two points at its ends, so you can certainly use those.

3. You may construct a circle centered at any existing point having a radius equal to the distance between any two existing points. In other words, you can set the size of the compass from any two points $A$ and $B$, and then you can move the point of the compass to another point $C$ without changing the setting and draw a circle of radius $AB$ about the point $C$¹. (Of course this includes drawing a circle given its center and a point on the edge--you use the center and the edge to set the compass size, and then you re-use the center point as the center of the circle.) As with the straightedge, there is no limit to the size of a circle that can be drawn, so the mathematical compass is better than any real one could be.

4. You may choose an arbitrary point on a line, circle, or on the plane. (And of course you can also choose a point not on a line or circle as in ``pick any point not on segment $AB$.'')