1. Introduction

Dynamic geometry software is now well established and there are many examples of specific implementations. One of the first of which was Cabri Geometry, one of the more recent is GeoGebra. At their heart this software allows the user to manipulate geometrical objects. First a configuration is created on screen in which, for example, relationships between points, lines and circles are established and then examined in a dynamic way by dragging.

As a specific example, in GeoGebra say, we might take three points A, B and C which are not co-linear and illustrate the unique circle through them using the button . We then draw lines AB and BC using twice. Next we find the midpoint of the segment AB, called D and that of BC, called E. There is a convenient "midpoint" button to facilitate this, shown as . Next we drop a perpendicular to AB through D, and another through BC through E, for which the tool "Perpendicular line", is indispensable. Where these two lines intersect, found using "Intersect two objects" , we have a point which it is not difficult to prove is the center of the circle. This has been labeled as O on the diagram in Figure 1.

What puts the "dynamic" into dynamic geometry, of course, is the ability to move the points A, B and C using the "Move" button , while the other relationships remain. Note that throughout this article you may either

1. click and manipulate the pictures or
2. double-click on the picture to obtain the dynamic construction in an interactive window. You can then save the construction on your own local machine for future use.

However, we need to consider carefully the consequences of what we have achieved. We could have used the "Line bisector" button to encapsulate the application of and . We could have further found the center of the circle directly by simply taking the circle and using the button , since in GeoGebra the "Midpoint or center" can be applied to other objects, not just a pair of points or a line segment. However, a circle is more usually given by a center point O and another point through which it would pass, , the distance between the two points being the radius r. If this is our only way to create a circle we would need to perform the above construction from three points to establish the position of O. This combined with one point A, B or C illustrates how the button actually works.

It is not clear at the outset in dynamic geometry how the buttons relate to each other, or that the possibility of catastrophic circularity can be avoided. This was something previous writers such as Euclid were at pains to avoid. Hence we ask "how does this button work?". Or, put another way, this article is one attempt to examine the dynamic geometry package GeoGebra from the point of view of Euclidean Constructions. Since GeoGebra is freely available to download, or use through a web browser as a Java applet, you are strongly encouraged to fully work through the examples in this paper for yourself.

By Euclidean Constructions we mean the classical topic of unmarked straightedge (also known as "ruler") and compasses construction. Euclid, in his Elements [Heath], took this approach, not because they were the only instruments of his day, but rather because he wanted to construct his geometrical theory using a minimum of assumptions. That is to say he wanted to assume only a small collection of self-evidently true axioms and derive, in a logically sound manner, the consequences of these, known as theorems. His assumptions were that it is possible to

1. draw a straight line from any point to any point,
2. extend a finite straight line indefinitely in a straight line,
3. draw a circle with any center and any diameter.

The first of these is implemented by the button and the first two combine to provide the button . We shall mostly consider indefinite straight lines. The third is the button . When Euclid talks of "any diameter" he means any previously constructed length. That is, he may use two existing points to open his compasses against. He does not mean any diameter we can imagine or perhaps define algebraically, eg 2^1/3.

Furthermore, he assumed that

4. all right angles are equal to one another, and
5. that if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which the angles are less than the two right angles.

This last assumption is sometimes known as the parallel postulate.

We note in passing that one strength of GeoGebra is the conjunction of algebraic and geometric views of mathematics. For example, one can type in the equation of a parabola as y = x², then place a point on the curve and drag it around. The tangent line can be illustrated using the button , and the equation of this line recovered from the algebra window. These features, while valuable, do not concern us here. We will only consider the purely geometrical operations.