Olive Oppong Transforming Mathematics with GSP 4 page 11 Chapter 1 Euclid s Construction Tools Euclid was a Greek mathematician who lived about three centuries before the common era c 300 BC His geometric treatise referred to as Euclid s Elements has been the most enduring and widely used mathematical work in the history of mathematics Euclid lays forth an axiomatic system in the Elements whereby most of the theorems of plane geometry can be derived through logical deduction from a small set of definitions postulates and axioms statements that are accepted without proof such as things equal to the same thing are equal to one another Euclid s Elements also contains many geometric construction problems that had to be carried out using very specific tools a collapsible compass and a straightedge Euclid s collapsible compass could be used to draw a circle or an arc of a circle from a given point as center of the circle passing through another given point that determines the radius of the circle or arc Because the compass was collapsible however it could not be used to transfer a given distance between two points onto a line that did not contain those two points as we can do with a modern compass whose radius can be fixed and then the compass moved to another point as center for duplicating the radius on a new line Euclid s straightedge was also different from a modern ruler in that it had no metric It could only be used for constructing line segments or lines passing through given points It could not be used for measuring distances between points or for duplicating a given distance by making marks on the straightedge Because of these restrictions even the most basic constructions were fairly complex For instance start with an arbitrary line segment and try to duplicate that line segment on a line not containing the line segment as in Figure 1 1 Olive Oppong Transforming Mathematics with GSP 4 page 12 B A C D Figure 1 1 Duplicate Segment AB on Line CD Following the strict rules of Euclidean construction this seemingly simple task requires the construction of 12 arcs or circles and 6 extra lines see Figure 1 3 The rules are as follows 1 You may draw a line or line segment between any two points 2 From any point you may draw an arc or circle of arbitrary radius 3 From any point you may draw an arc or circle that passes through any other point 4 From any point you may draw an arc or circle to intersect any line or another arc or circle that has a different center 5 You may construct points at the intersections of any lines line segments circles or arcs Given these strict rules one approach to duplicating the segment AB on the line CD at point C is to construct a parallelogram with sides CA and AB However in order to construct a line through a point parallel to a given line or segment following the above rules requires the construction of pairs of perpendicular lines why It turns out that the construction of a line perpendicular to a given line through a given point is the basic building block for most of the possible constructions using Euclidean construction tools Figure 1 2 illustrates the construction of a line perpendicular to line AB through the point C not on AB Olive Oppong Transforming Mathematics with GSP 4 page 13 c3 G c2 EA F B c1 C H Figure 1 2 Construction of a Perpendicular to a Line Using Euclidean Construction Tools The steps in the construction of the perpendicular to line AB through C are as follows 1 Draw a circle c1 center C to intersect line AB in two points E and F 2 Draw a circle c2 center E passing through point F 3 Draw a circle c3 center F passing through point E 4 Draw a line through the intersection points G and H of circles c2 and c3 This line passes through point C and is perpendicular to line AB Activity 1 1 Define perpendicular lines in your own words Provide a rationale for why the four construction steps above produce a line perpendicular to line AB through point C Using similar steps construct the line through C that is perpendicular to the line GH Will this line be parallel to line AB If so why Olive Oppong Transforming Mathematics with GSP 4 page 14 B A V C X D Figure 1 3 Duplicating Segment AB on Line CD at Point C Using Euclidean Construction Tools Figure 1 3 illustrates the complexity of the construction steps needed to duplicate a segment following the Euclidean construction rules Segment AB is congruent to segment CX on line CD The parallelogram ABVC was constructed through several applications of the steps needed to construct a line parallel to a given line or segment through a point not on that line see assignment 1 1 above Twelve construction circles and 6 construction lines were needed to construct segment CX on line CD If we were to use a modern non collapsible compass we could simply set the radius of our compass using segment AB and with the point center of the compass at point C draw an arc or circle with radius AB to intersect line CD at point X basically two simple steps Most if not all high school geometry texts used in our schools Olive Oppong Transforming Mathematics with GSP 4 page 15 today allow the use of a modern compass in construction problems The ancient Greeks also used their physical compasses to mark off distances as with a modern non collapsible compass Euclid s rules of construction were mathematical rules that avoided the necessary error that could creep into a construction when using the compass to duplicate a length In practice of course error could be introduced when drawing arcs or circles with a compass the radius may not stay fixed during the drawing process Euclid s constructions were geometric exercises to train the mind rather than engineering problems and so the mathematical purity was what mattered In the following construction problems you may use a modern non collapsible compass to duplicate lengths of segments For each construction problem first write a definition in your own words for the object that you are to construct then carry out the construction using only a non collapsible compass and straight edge Activity 1 2 Construction Problems 1 Construct the perpendicular bisector of a given segment 2 Construct the angle bisector of a given angle 3 Construct a line parallel to a given line through a given point not on the line 4 Construct the midpoint of a given segment 5 From a point not on a given line construct the shortest segment from the point to the line 6 Construct an equilateral triangle given one