3. Theorems

Those buttons which correspond to theorems are shown in Table 1. Notice that many of the buttons are overloaded in the sense that they can apply to different combinations of objects. For example, the "Midpoint or center" can take either (i) 2 points, (ii) a line segment, (iii) a circle, or (iv) a conic and returns a point. Each of these four possibilities is a separate theorem, requiring a construction of its own.

Table 1. Buttons which are Euclidean theorems

Theorem/construction	Button	Applies to	Result
Line bisector		Two points, (or line segment)	Line
Perpendicular line		Line and point	Line
Parallel line		Line and point	Line
Midpoint or center		(i) 2 points, (ii) circle, (iii) conic	Point
Angular bisector		Three points or two lines	Line(s)
Circle with center and radius		Radius must only be distance between existing points	Circle
Semicircle through two points		2 points	Circle
Circle through three points		Three points, not co-linear	Circle
Mirror object at point		Any object + point	Same object
Mirror object at line		Any object + line	Same object
Tangents		Point + circle/conic	Line(s)
Polar or diameter line		Point + circle/conic	Line

Most of the theorems shown in Table 1 are standard exercises in construction when taken in the order shown above. You are encouraged to work them out for yourself and hence I will make no further comment on how to perform them. Those involving conics are more advanced and so these, together with the status of as an axiom or theorem is something to which we shall return in Section 4.

In the case of the Circle with center and radius, , GeoGebra expects the user to type in an algebraic distance. Hence the status of this button as a purely geometric theorem is questionable. In order to include this button we would need to include the "Distance tool", or use GeoGebra's "Segment between two points" which returns the length of the segment.

In addition it might be advantageous to build a basic arithmetic system by identifying the length of a line segment with a number, starting with an arbitrary agreed unit. Then, in a systematic way, to construct the geometric counterparts of the arithmetic operations such as addition, multiplication, and so on. It is not at all clear which operations and hence numbers are constructible in this way. This combination of algebra and geometry is a classical topic and the basic geometric constructions for addition, multiplication are given in [Bryant & Sangwin].

It is a famous result attributed to Mascheroni that any straightedge and compass construction that can be performed with straightedge and compass can be performed with compass only. See, eg [Kostovskii]. Hence the straight line button is a theorem, rather than an axiom. While it is not possible to draw the resulting straight line with compasses, it is possible to draw segments with linkworks.

A planar linkwork is a physical mechanism comprising collection of individual ridged linkages. These linkages are connected together by pivots which allow the links to rotate around an axis perpendicular to the plane. Within such linkworks the points which represent the pivots are constrained to move on circles. Hence linkworks provide a model in which compass-only constructions may be investigated. The simplest linkwork was invented by James Watt and the most famous linkwork for generating a straight line segment is known as Peaucellier's linkage, named for Charles-Nicolas Peaucellier.

To construct a straight line segment, start with any two points O and A. It is trivial to construct another point which is greater than twice but less than three times the distance between OA; say 5/2 this distance. A new point Q can be placed arbitrarily on the screen between these distances and a circle c₁ constructed centered at O through Q. Next, construct a circle centered at A through O and place a free point B on it. Construct a circle c₂, centered at B through A. Let C and D be the intersections of c₁ and c₂. Place circles centered at C and D, both through B. The intersection of these two circles will be B and an additional point P. The locus of P as B moves is a straight line. In GeoGebra the locus can either be illustrated using , or by typing locus[P,B] into the input line. See Figure 2.

Note however that this constructs a straight line segment, not a straight line segment through two given points. However this segment is perpendicular to OA, and intersects OA a distance [(OQ² − OA²) / (2OA)] from O. Furthermore while the points on this segment do exist as a set of intersection points there is no way to use them in later constructions. Hence there is an important and interesting distinction between intersection points which exist in an abstract sense and those which arise from a concrete construction and which can actually be subsequently used.

In Figure 2 it is not immediately obvious that the circles represent a linkage. To operate this construction move the point B. Here the short line segments have deliberately been omitted and you might like to look at the video clip below and work out for yourself where the links go.

An alternative form of this linkage is also shown below.

For more information on linkages see, for example, [Bryant & Sangwin].