cabriplusmanual.pdf

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CabriGeometry II Plus

UserManual


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Welcome!

Welcome to the world of dynamic geometry!

Cabri Geometry was created in the eighties, in the research laboratoriesof CNRS (Centre National De Recherche Scientifique) and at the JosephFourier University in Grenoble. Fifteen years later, there are more than tenmillion users of the software, using Cabri Geometry on personal computers

operating under Mac

OS and Windows

, also on the TI-92, TI-92 Plus,TI Voyage

200, TI-89 and TI-83 Plus calculators from Texas Instruments

.Cabri Geometry is currently developed and distributed by Cabrilog, acompany which was founded in March 2000 by Jean-Marie Laborde, directorof research at CNRS, and virtual father to the line of Cabri babies.

Computer-assisted construction of geometrical diagrams brings a newdimension to the classical method of construction using paper, pencil, rulerand compasses. In fact, once a diagram has been constructed, it can befreely manipulated. Conjectures can formulated and tested. Measurementsand calculations can be made. Part of the diagram can be erased, or thewhole thing redrawn from the beginning... Once the diagram is complete,the intermediate constructions can be hidden, colour or broken lines canbe added, as can text. The diagram is then ready for distribution over theInternet, or for incorporation into another document.

Cabri Geometry II Plus is a new version of the Cabri Geometry IIsoftware. It has many new features which make it even more powerful andeasy to use. In addition, this version has had the bugs of the old versioncorrected, and includes much of the functionality requested by its users.

This manual is in three sections. Part one Discovery - IntermediateTutorial is designed for new users, and suggests activities at secondary

school level. Part two Moving On - Advanced Tutorial suggests moreadvanced activities for A-level or undergraduate work. Finally, Part threeReference Section is the complete reference document for the software.

The various activities in the first two parts are largely independent of eachother. The reader is invited to duplicate the detailed construction methods,then try the listed exercises. The exercises marked with an asterisk (*) aremore difficult.

Before using Cabri Geometry for the first time, we recommend that newusers read the chapter in the introduction headed Starting Out - BasicTutorial, to familiarise themselves with the Cabri Geometry interface, and


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the conventions for using the mouse. However, experience shows that thelearning curve for Cabri Geometry is steep, and that in class, students arealready doing geometry within half an hour of loading the software.

Our website, www.cabri.com, will give you access to the latest updatesand product news, and in particular the new versions of this manual.The website also has links to dozens of Internet pages, and informationconcerning books about geometry and Cabri Geometry.

All the Cabrilog team wish you many fascinating hours of constructions,explorations and discoveries.

2002 Cabrilog SAS

Author: Eric BainvilleTranslation: Sandra HoathLatest update: 19th September 2002New versions: www.cabri.comMistakes: [email protected]

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mailto:[email protected]://www.cabri.com/en/http://www.cabri.com/fr/


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Contents

Chapter 1 Starting Out - Basic Tutorial 5

Part One Discovery - Intermediate Tutorial

Chapter 2 The Euler Line 15

Chapter 3 Hunt the point 21

Chapter 4 The Varignon quadrilateral 24

Part Two Moving On - Advanced Tutorial

Chapter 5 Pedal triangles 30

Chapter 6 Functions 35

Chapter 7 Tessellations I 40

Chapter 8 Tessellations II 46

Part Three Reference Section

Chapter 9 Objects and Tools 52

Chapter 10 Investigative Tools 67

Chapter 11 Attributes 70

Chapter 12 Preferences and Customisation 74

Chapter 13 User Interface 79

Chapter 14 Exporting and Printing 95


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Chapter 1

StartingOut -BasicTutorial

1.1 Philosophy

The philosophy behind Cabri Geometry is to provide the greatest flexibilityin interaction (keyboard, mouse...) between the user and the software. Ineach case the software behaves in the way that the user expects: on theone hand by respecting the industry standards, and on the other hand byfollowing the most plausible mathematical route.

A Cabri Geometry document consists of a diagram which is free to bedrawn anywhere on a virtual 1m square sheet of paper. A diagram consistsof standard geometrical objects such as points, lines, circles... and also

numbers, text, formulae...

A document can also contain construction macros, which enable interme-diate constructions of a diagram to be easily reproduced, and which extendsthe functionality of the software. Cabri Geometry allows several documentsto be open simultaneously.


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1.2 User Interface

Figure 1.1 The Cabri Geometry window and its different regions.

Figure 1.1 shows Cabri Geometrys main window and its different regions.When Cabri Geometry is first loaded, the Attributes toolbar, the helpwindow and the text window are not displayed.

The title bar displays the diagrams filename, or Figure 1, 2... if thediagram has not yet been saved.

The menu bar enables the user to draw on the applications commands,which correspond to the usual software commands.

In the rest of this manual, we shall designate the command Actionfrom the Menu menu by [Menu]Action. For example, [File]Save As...

designates Save As... from the File menu.

The toolbar displays the tools which enable the diagram to be created andmodified. It consists of several toolboxes, each of which displays one toolfrom the toolbox as an icon on the bar. The active tool is shown as a pressedbutton, with a white background. The other tools are shown as unpressedbuttons with a grey background. A brief, single click on a button activatesthe corresponding tool. Click-and-hold on the button opens the toolbox as adrop-down menu, and by dragging to another tool it can be made the activeone which is then displayed as the icon for the toolbox.

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The toolbar can be freely modified by the user, and ultimately lockedinto a configuration for use by a class. (See the chapter: Preferences andCustomisation in the Reference part of this manual.)

Figure 1.2 Cabri Geometrys default toolbar, with the names of the varioustoolboxes.

In the remainder of this manual, we shall designate the Tool tool fromthe Toolbox toolbox by [Toolbox]Tool, and show the corresponding iconin the margin. (Some of the labels which are too long for the margin havebeen abbreviated.) For example, [lines]Ray represents the Ray tool fromthe lines toolbox.

The toolbar icons can be displayed in large or small format. To change thesize, move the cursor to a position to the right of the last tool shown on thetoolbar, and click with the right mouse button (right-click).

The status bar gives a permanent indication of the active tool.

The attributes bar enables the attributes of different objects to bechanged: colour, style, size... It is activated by the command [Options]-Show Attributes, and de-activated by [Options]Hide Attributes. Thefunction key F9 can be used for the same purpose.

The help window provides outline help regarding the active tool. Itdisplays the anticipated required objects, and what will be constructed. Itis activated/de-activated by the F1 key.

The history window contains a description of the diagram in text form.It lists all the objects that have been constructed, and the constructionmethods used. It is opened with the command [Options]Show historywindow, and closed with [Options]Hide history window. It can also betoggled by F10.

Finally, the drawing area shows part of the total area that is available.It is in this drawing area that geometrical constructions are carried out.

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1.3 Using the Mouse

Most of the software functions are controlled by mouse operations. Theoperations required are: using the mouse to move the cursor, pressing ona button, releasing the button. A brief press-and-release sequence is calleda click. A rapid press-release, press-release sequence is a double-click.The sequence press-move-release is called drag-and-drop, if it is used tomove an object, or click-and-drag when it is used to stretch out a selectionrectangle. In the absence of any other indication, the button referred to isthe main mouse button, usually the left mouse button.

If a modifier key: Alt, or Ctrl, is inserted in a sequence, it is possiblethat the action will be changed. Ctrl-click is a click which is carried outwhile the Ctrl key is held down. Similarly for other combinations.

When the mouse is used to move the cursor across the drawing area, thesoftware informs us in three ways of the anticipated action of a click or adrag-and-drop:

the shape of the cursor,

a pop-up message displayed alongside the cursor,

a partial display of the object being constructed.

Depending on the construction, the pop-up message and the partial object

may not be displayed.

The different cursors are as follows:

An existing object can be selected.

An existing object can be selected, or moved, or used in a construc-tion.

Appears when an existing object has been clicked on to select it, orto use it in a construction.

Several selections are possible for the ob jects under the cursor. A

click causes a menu to appear which enables the precise object to beselected from a pop-up list.

Appears while moving an object.

The cursor is in an unused portion of the sheet, and here a rectangu-lar selection area can be made using click-and-drag.

Indicates the pan mode for moving the visible area of the sheet. Thismode can be entered at any time by holding down the Ctrl key. Inthis mode, drag-and-drop slides the worksheet across the window.

Appears as the worksheet is panned.

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Indicates that a click will create a new independent, movable pointon the sheet.

Indicates that a click will create a new independent, movable point

on an existing object, or a new point at the intersection of twoexisting objects.

Indicates that a click will fill the object under the cursor with thecurrent colour.

Indicates that a click will change the attribute (for example thecolour, style, thickness...) of the object under the cursor.

1.4 First Construction

As an illustration for this chapter Starting Out - Basic Tutorial, we shallconstruct a square, given one of the diagonals.

When Cabri Geometry is loaded, a new, blank, virtual drawing sheet iscreated, and the user can immediately start a construction.

We shall first construct the segment which will be the diagonal of thesquare. The [lines]Segment tool must be activated by clicking on thelines icon and then holding down the button of the mouse to open up thetoolbox. Move the cursor to the Segment tool and release the mouse buttonto activate it.

Figure 1.3 Selection of the tool[lines]Segment.

Figure 1.4 Construction of thefirst point. A pre-image of the finalsegment moves with the cursor untilthe second point has been selected.

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Figure 1.5 The segment is completeafter the selection of the second point.

The [lines]Segment tool remainsactive, enabling the user to constructanother segment.

Now move the cursor across the drawing area: it will assume the form. Clicking creates the first point. Continue to move the cursor across

the drawing area. A segment will extend from the first point to the cursor,

showing where the segment will be created. The second point is also createdby clicking. Our drawing now contains two points and one line segment.

To construct the square, we first need the circle with this segment asits diameter. The centre of the circle is the midpoint of the segment. Toconstruct this midpoint, activate the [constructions]Midpoint tool thenmove the cursor over the segment. The pop-up message Midpoint of thissegment is displayed alongside the cursor, whose shape changes to . Themidpoint is marked on the segment by clicking.

Figure 1.6 Construction of themidpoint of a segment.

After this, activate the [curves]Circle tool and move the cursor near tothe midpoint just drawn. The pop-up message This point as centre is thendisplayed. The [curves]Circle tool requires the selection of a point as thecentre of the circle, so click to select it. As the cursor is moved following thisselection, a circle is displayed; so move the cursor near to one end of the line

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segment, when Cabri Geometry displays Through this point. By clicking,the circle is completed.

Figure 1.7 Construction of the circle with the given segment as diameter.

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Use the [manipulation]Pointer tool to change the diagram. The onlymovable points of the diagram are the endpoints of the line segment. Whenthe cursor is moved over one of them, its shape becomes and the pop-up

message This point is displayed. The point can then be moved by drag-and-drop and the entire diagram is automatically updated: the segment isredrawn, the midpoint moves to follow suit, as does the circle.

Figure 1.8 Construction of theperpendicular bisector of the linesegment, to determine the otherdiagonal of the square.

To construct the square we need the other diagonal, which is the diameterof the circle, perpendicular to the original segment. We shall construct theperpendicular bisector of the segment: a line, perpendicular to the seg-ment, through its midpoint. Activate the [constructions]Perpendicularbisector tool, and then select the segment by clicking on it. The perpendic-ular bisector is constructed.

Finally, to construct the square, activate the [lines]Polygon tool. Thistool expects the selection of a sequence of points to define the vertices. Thesequence is terminated by selecting for a second time the initial point ofthe sequence, or by double-clicking to select the last point of the sequence.

The two points of intersection of the circle with the perpendicular bisectorare not actually constructed: Cabri Geometry enables them to be selectedimplicitly as they are used.

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Figure 1.9 Construction of thesquare, using implicit selection of thepoints of intersection of the circle andthe perpendicular bisector.

In other words, select one endpoint of the segment (pop-up message Thispoint) as the first vertex of the polygon, then move the cursor to one of thepoints of intersection of the circle and the perpendicular bisector. A pop-upmessage This point of intersection is displayed to show that a mouse-clickwill construct the point of intersection and select it as the next vertex of thepolygon, so select it. Follow this with the selection of the other endpoint

of the line segment, the second point of intersection of the circle and theperpendicular bisector, and finally the initial vertex. The square appears.

Figure 1.10 Your first Cabri Geometry construction!

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PartOne

Discovery - IntermediateTutorial


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Chapter 2

TheEulerLine

We shall construct a general triangle ABC, then its three medians. Theseare the lines that join a vertex to the midpoint of the opposite side. We

shall then construct the three altitudes of the triangle: the lines througheach vertex in turn, perpendicular to the opposite side. Finally we shallconstruct the three perpendicular bisectors of the sides of the triangle: linesperpendicular to each side, through the midpoint of the side.

It is a well-known fact that the three altitudes, the three medians and thethree perpendicular bisectors are concurrent, and these points of concurrencylie on a straight line, called the Euler1 line of the triangle.

To construct a triangle, choose the [lines]Triangle tool. For informationon how to use the toolbar, consult the Chapter Starting Out - Basic

Tutorial in the Introduction.Once the [lines]Triangle tool is active, select three new points in the

drawing area, by clicking in an empty space. These points can be labelledimmediately after their creation on the fly simply by typing their labelon the keyboard. Once the triangle has been constructed, these labels canbe moved around the points to place them, for example, outside the triangle.

A

B

CFigure 2.1 Triangle ABC isconstructed using the [lines]Triangletool. The vertices are labelled on thefly by typing a letter at the time they

are created.

1Leonard Euler, 1707-1783


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To move an objects name, the [manipulation]Pointer tool must beactive. Drag the name by positioning the cursor over it, then holdingdown the mouse button while dragging the mouse to move the name to the

desired location. To change the name of an object, activate the [text andsymbols]Label tool then select the name, at which point an editing windowwill appear.

The midpoints are constructed, using the [constructions]Midpoint tool.To construct the midpoint of AB, select in turn A then B. The midpointof a segment can be constructed equally well by selecting the segment itself.The new point can be named on the fly, say C. The midpoints of the othersides are constructed in the same manner: A on BC and B on CA.

A

B

C

A'B'

C'A

B

C

A'

B'

C'

Figure 2.2 [Left]. The midpoints are constructed with the [constructions]-Midpoint tool, which accepts as arguments two points, a segment, or the side ofa polygon. [Right]. The medians are constructed with the [lines]Line tool, andtheir colour is changed with the [attributes]Colour tool.

The [manipulation]Pointer tool enables the independent, movableobjects of a construction to be moved freely. In this case, the three points A,B and C are the independent, movable objects. The entire construction isupdated automatically as soon as any of them is moved. It is thus possibleto explore all the various configurations of a construction. To determinewhich are the movable objects of a diagram, activate the [manipulation]-

Pointer tool, then click and hold on an empty part of the drawing areawhich, after a short delay, causes the movable objects to be displayed asmarquees (also known as marching ants).

The [lines]Line tool enables the three medians to be constructed. For theline AA, click successively on A then A. The [attributes]Colour tool isused to change the line colour. Select the colour from the palette, by clickingon it, and then click on the object to be coloured.

Activate the [points]Point tool, and then select the point of intersectionof the three medians. Cabri Geometry tries to construct the point of

intersection of two lines but, since there is an ambiguity (there are three

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concurrent lines to choose from) a menu appears enabling the user to selectwhich two lines to use to construct the point. As the cursor is moved downthe list of options, the corresponding lines in the diagram are highlighted.

Label the point of intersection of the medians G.

A

B

C

A'

B'

C'

G

Figure 2.3 Construction of the point of intersection of the medians, and theresolution of the ambiguities of a selection.

The altitudes of the triangle are constructed with the [constructions]-Perpendicular line tool. This tool constructs the unique line which isperpendicular to a given direction, through a given point. Therefore, select apoint and: a line, a segment, a ray... The order of selection is not importanthere. To construct the altitude through A, select A then the side BC. The

altitudes through B and C are constructed similarly. In the same way as forthe medians, choose a colour for the altitudes, and construct their point ofintersection H.

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A

B

C

A'

B'

C'

G

H

A

B

C

A'

B'

C'

G

HO

Figure 2.4 [Left]. The altitudes are constructed using the [constructions]-

Perpendicular line tool. [Right]. Finally, the perpendicular bisectors areconstructed using the [constructions]Perpendicular bisectortool.

The [constructions]Perpendicular bisector tool is used for theconstruction of the perpendicular bisector of a segment, or the line which isequidistant from two points. Label O, the point of intersection of the threeperpendicular bisectors.

The [properties]Collinear? tool enables us to check numericallywhether the three points O, H and G are collinear. By selecting each ofthese points in turn, then clicking somewhere on the drawing area, the

answer is displayed. The answer is a sentence saying whether or not thepoints are collinear. If the independent points of the diagram are moved,this text is updated at the same time as the other parts of the diagram.

Points are collinear

A

B

C

A'B'

C'

G

HO


A

B

C

A'B'

C'

G

HO

Figure 2.5 [Left]. Numerical collinearity check for the three points O, Hand G. The[properties]Collinear? tool creates a text message Points arecollinear or ...not collinear. [Right]. The Euler line of the triangle, shownclearly by its increased thickness, as changed by the [attributes]Thick tool.

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You should be aware that the check is numerical. In other words it isbased on the 16-digit coordinates calculated by the software for each of thethree points. Rounding errors can give a false result, but it is very unlikely

for simple diagrams. The result is in no way a mathematical proof.

The Euler line of the triangle is constructed with the [lines]Line tool,through the three points O, H and G, by selecting, for example, O and H.The [attributes]Thick tool is used to distinguish this line.

When the shape of the triangle is changed by moving the relative positionof the vertices, it is apparent that G is always between O and H, and alsothat its relative position on the line segment does not change. Supposewe check this by measuring the lengths of GO and GH. Activate the[measurement]Distance and length tool. This tool measures thedistance between two points, or the length of a line segment, depending onthe object selected. Select G and then O: the distance from G to O appears,measured in cm. Do the same for G and H. Once the measurement hasbeen taken, the corresponding text message can be edited by adding GO= infront of the number, for example.


GO = 1.10 cm

GH = 2.20 cm

A

B

C

A'B'

C'

G

HO


GO = 1.10 cm

GH = 2.20 cm

GH/GO = 2.0000000

A

B

C

A'B'

C'

G

HO

Figure 2.6 [Left]. The [measurement]Distance and length tool hasbeen used to find the lengths of GO and GH. [Right]. Using the calculator [measurement]Calculate to display the ratio GH/GO and show that it isalways equal to 2.

By making changes to the original triangle, GH can be seen to be alwaystwice the length of GO. Let us calculate the ratio GH/GO to check this.Use the [measurement]Calculate tool. Select the text message givingthe distance GH, then the operator /, and finally the text message givingGO. Click on the = key to get the result, which can be dragged to thedrawing area. When a number is selected ([manipulation]Pointer tool),the number of digits displayed can be increased or decreased by means of the

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+ and - keys. In this way, the ratio can be displayed with a dozen digits, toshow that it is always equal to 2.

Exercise1 Add to the diagram the circumscribed circle, centre O, passingthrough A, B and C. Use the [curves]Circle tool.

Exercise 2 Next, add the nine-point circle for the triangle. This is thecircle whose centre is at the midpoint of OH, and which passes through themidpoints of the sides: A, B and C, the foot of each altitude, and themidpoint of each of the line segments HA, HB and HC.

AB

C

A'B'

C'

G

HO

Figure 2.7 The final diagram, showing the triangle with its circumscribedcircle and nine-point circle.

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Chapter 3

Hunt the point

In this chapter, we shall illustrate an activity which shows the possibilitiesfor exploration that are provided by Cabri Geometry. Starting from three

given points A, B, C, we shall look for any points M such that

M A +

M B +

M C =

0 .

First, however, we shall construct four points in random positions, usingthe [points]Point tool, labelling them A, B, C, and M on the fly, i.e. bytyping the appropriate letter immediately after each point has been created.

Cabri Geometry allows vectors to be used. As we are in an affine plane,we shall manipulate the vectors as their line segment representations, whichare suitably labelled and given an arrow to indicate their sense.

We now need to construct the vectorM A using the [lines]Vector tool, by

selecting first M, then A. This line segment representation of the vector hasits origin at M. Do the same for

M B, and

M C.

Next, construct the resultant vector ofM A +

M B using the

[constructions]Sum of two vectors tool. Click first on the two vec-tors and then on the origin for the resultant, choosing M here. Label N, thefurther extremity of the vector. Finally construct the resultant of the threevectors, with M as its origin, in the same way, adding vecMN (which equalsM A +

M B) and

M C. Label the further extremity of this vector: P.


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M

A

B

CM

A

B

C

N

P

Figure 3.1 [Left]. Starting from any three points: A, B, C, and a further

point M, the line segments representing the vectors MA, M B and MC aredrawn. [Right].

MN =

M A +

M B, and

M P =

MA +

MB +

M C are

constructed, using the [constructions]Sum of two vectors tool.

We can now look for the solution to the problem diagrammatically. To dothis, activate the [manipulation]Pointer tool and move the point M. Theresultant of the three vectors is continually updated as M is moved aroundthe drawing area.

The magnitude and direction ofM P can be seen to depend on the

position of M relative to the points A, B and C. In this way, the following

observations and conjectures (among others) can be made:

There is only one position of M for which the resultant of the threevectors is null: the problem has a unique solution. The solution pointis inside the triangle ABC.

The quadrilateral MANB is a parallelogram.

The quadrilateral MCPN is a parallelogram.

For a zero resultant vector, the vectorsM N and

M C must be

collinear, and in addition they must have the same magnitude butopposite sense, in other words they must be equal and oppositevectors.

M P always passes through the same point and this point is thesolution to the problem.

The position of point P is dependent on M. Based on this fact, onecan define a transformation, and the solution to the problem is theinvariant point under the transformation.

According to the various observations made, the investigation can be takenin any of several directions. In a class situation, the students can decide

for themselves which direction to take, depending on their observations and

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previous experience. Some investigations take longer to achieve than others.For example here, studying the transformation (the last point mentionedabove) could be a more delicate matter.

Suppose for example, that the observation has been made that vectorsM N and

M C must have opposite directions. Another question then arises:

for which positions of M are these two vectors collinear? Move M in sucha way that the two vectors are collinear. It can be seen that M must lieon a straight line, and that this line passes through C and the midpoint ofAB. The line is therefore the median of the triangle through C. Since Mis equally dependent on A, B and C, it can be seen that M must also lieon the other two medians, and the required point is therefore the point ofintersection of the three medians.

As a class activity, the students could continue by developing a construc-tion of the solution point, and a proof of the hypothesis resulting from theinvestigation.

The illustrative power of a dynamic construction is much higher than thatof a static diagram drawn on a sheet of paper. In fact, it is sufficient tomanipulate the diagram to check the construction in a large number of cases.A construction which remains valid after a diagram has been altered will becorrect in the great majority of cases.

To use it to best effect in class, it is a good idea first to raise the followingpoints with the students (among others):

Is a dynamic construction that is visually correct actually correct?

Is a correct dynamic construction an answer to the question?

When can a mathematical argument be given the status of proof?

What is missing from a dynamic construction to make it a proof?

Must a proof be based on the procedure used to draw the diagram?

Exercise 3 Extend the problem to four points, by finding those points M,such that

M A + M B + M C+ M D = 0 .

Exercise* 4 List all the paths of exploration and proofs needed forthe initial problem (three points) which are available to a student studyingA-level.

Exercise* 5 Investigate and construct the points M which minimise thesum of the distances to three points: MA + MB + MC. The solution is theFermat1 point of triangle ABC.

1Pierre Simon de Fermat, 1601-1665

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Chapter 4

TheVarignonquadrilateral

In this chapter, we shall present a number of constructions based onVarignons1 Theorem.

First, construct any quadrilateral ABCD. Activate the [lines]Polygontool, then select four points and label them on the fly: A, B, C, D. To finishoff the polygon, reselect A after constructing D.

Next, construct the midpoints: P of AB, Q of BC, R of CD and S ofDA using the [constructions]Midpoint tool. This tool expects the user toselect A then B to construct the midpoint of AB. It is equally possible toselect the segment AB if this already exists: either as a line segment, or asthe side of a polygon, which is the case here.

Finally, construct the quadrilateral PQRS, using the [lines]Polygon tool.

By altering the diagram, using the [manipulation]Pointer tool, it canbe seen that PQRS always seems to be a parallelogram. We shall nowask Cabri Geometry to pronounce on whether or not the lines P Q andRS are parallel, then similarly for P S and QR, using the [properties]-Parallel? tool. Select the side P Q, then RS, and a text message willappear, confirming that the two sides are indeed parallel. Be aware thathere also, the check is numerical, and it is possible in very complex diagramsfor the result to be incorrect. Check in the same way that P S and QR areparallel.

1Pierre Varignon, 1654-1722


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A

B

C

D

P

S

R

Q

A

B

C

D

P

S

R

Q

I

Figure 4.1 [Left]. Starting from any quadrilateral ABCD, the quadrilateral

PQRS is constructed with vertices at the midpoints of the sides of ABCD.[Right]. Construction of the diagonals of PQRS, and the demonstration thatthey bisect each other.

So now construct the two diagonals P R and QS, using the [lines]-Segment tool, and their point of intersection I using the [points]Pointtool. There are several ways in which it can be demonstrated that I is themidpoint of both P R and QS, and that PQRS is therefore a parallelogram.For example, one can use centres of mass. P can be considered as thecentre of mass of two particles of equal mass at A and B {(A, 1), (B, 1)}.Similarly R is the centre of mass of particles of equal mass at C and

D {(C, 1), (D, 1)}. Thus the midpoint of P R is the centre of mass of{(A, 1), (B, 1), (C, 1), (D, 1)}. The midpoint ofQS is the same thing. Hencethe two midpoints coincide: at the point of intersection I.

Varignons theorem is as follows:

Varignons Theorem. The quadrilateral PQRS whose vertices are themidpoints of the sides of any quadrilateral ABCD, is a parallelogram whosearea is half that of ABCD.

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A

B

C

D

P

S

R

Q

Figure 4.2 The constructionto establish the second part of the

theorem.

Exercise 6 We have already demonstrated the first part of the theorem.Now show that the second part of the theorem is true. Hint: use thediagram of Figure 4.2.

Leaving A, B and C alone, move D so that PQRS appears to be arectangle. Since we already know that PQRS is a parallelogram, it issufficient to show that one of its angles is a right angle. So, measure theangle at P, using the [measurement]Angle tool. This tool expects theuser to select three points, the vertex of the angle being the second point.Here, for example, one should select S, P (the vertex of the angle) and Q.

102,7

A

BC

D

P

S

R

Q

Figure 4.3 Measuring angle P ofthe parallelogram.

The [measurement]Angle tool can also be used to give the size of anangle which has previously been marked with the [text and symbols]MarkAngle tool. This tool also expects three points to be selected, in the sameorder as for [measurement]Angle.

By moving D so that PQRS is a rectangle, it can be seen that there isan infinity of solutions, as long as D lies on one straight line. In fact, if the

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diagonals AC and BD of the quadrilateral ABCD are drawn, it can be seenthat the sides of PQRS are parallel to them, and hence PQRS is a rectangleif and only if AC and BD are perpendicular.

To ensure that PQRS is always a rectangle, we need to redefine theposition of D. Draw the line AC with the [line]Line tool by selecting A andC, then draw the perpendicular to this line which passes through B, usingthe [constructions]Perpendicular line tool, selecting B and the line AC.

D is currently an independent, movable point of the diagram. We shallmodify this so that it becomes a point which is constrained to lie on theperpendicular to AC. Activate the tool [constructions]Redefine Object,then select D. A menu appears listing the various options for redefining D.Choose Point on an object, then select any point on the perpendicular. Dmoves to this point, and thereafter is constrained to be on the designatedline.

Redefinition as a powerful investigative tool, which enables the user toincrease or decrease the number of degrees of freedom in a diagram withouthaving to redraw it from scratch.

Figure 4.4 Point D is nowredefined so that PQRS is always arectangle. D still has one degree offreedom, being able to move along aline.

90,0

A

BC

D

Q

R

S

P

Exercise 7 Find a necessary and sufficient condition that PQRS is

a square. Redefine D again, so that the construction will only producesquares.

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A

B

C

D

P

S

R

Q

Figure 4.5 Here, D has no degrees of freedom at all, and PQRS is always a

square.

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PartTwo

MovingOn-AdvancedTutorial

A

B

C


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Chapter 5

Pedal triangles

Use the [points]Point tool to start with three points, A, B, C, anywhereon the drawing area. First construct the straight lines AB, BC and CA,

using the [lines]Line tool. Create a fourth point M, anywhere on the plane,and the orthogonal projections of M: C, A and B, respectively, on theselines. These points are constructed by first creating the perpendicularsthrough M to each of the lines in turn, using the [constructions]-Perpendicular line tool. Use the [points]Point tool to pick up thepoint of intersection of each perpendicular with its corresponding line. The[points]Point tool constructs implicitly the points of intersection of twoobjects. It merely requires the cursor to be placed close to an intersection,when Cabri Geometry displays the message Point at this intersection or,in an ambiguous case Intersection of... followed by a menu list.

The three points A

, B

and C

define a triangle which can be drawnusing the [lines]Triangle tool. It is called a pedal triangle. The interiorof the triangle can be coloured, using the [attributes]Fill tool. The pointof interest here is the area of the triangle with regard to the position ofM. The area of the triangle is measured, using the [measurements]Areatool. The resulting value is a geometrical area, taking no account of theorientation of the triangle. The measurement is given in cm2 and can beplaced anywhere on the drawing area. By clicking the right mouse button,a shortcut menu appears, with the option to change to the algebraic area,the sign of which depends on the orientation of the triangle.


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Area = 12.19 cm

AB

C

M

A'

C'

B'Figure 5.1 The pedal triangle forM, and its area.

We shall consider how the area of ABC varies, depending on the positionof M. There are several possible strategies for this. For example, activatethe [text and symbols]Trace On/Off tool (which requires selection of theobject to be traced, M here so click on it). Now move M while attemptingto keep the area of ABC constant. Successive positions of M are displayedon the screen, giving the appearance of a contour line for equal values of thearea. Another strategy could be to use the locus of points on a grid to drawa visual representation of the area of ABC for a large number of positionsof M.

Here, we shall use this latter strategy, and draw the circle, centre M,which has an area proportional to that of ABC for a large number ofpositions of M. To do this, it is necessary first to calculate the radius of thecircle, proportional to the square root of the area of the triangle. Activatethe [measurements]Calculate tool, and enter the expression sqrt( thenselect the number displaying the area of the triangle to insert that into theexpression, which becomes sqrt(a. Now close the bracket, and divide by 10to avoid having a circle which is too large. The expression in the calculatoris now sqrt(a)/10. Evaluate this by clicking on the = button, then drag theanswer to an appropriate position on the sheet.

To draw a circle, centre M, using the radius we have just calculated,activate the tool [constructions]Compass. Select the number, then thepoint M. The circle, centre M, with the required radius appears. We cannow see the changes in the area of the circle surrounding M, as the point ismoved.

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Figure 5.2 A circle is drawn,centre M, with area proportional to

that of A

B

C

.

r = 0.35 cm

Area = 12.19 cm

AB

C

M

A'

C'

B'

We shall now define a grid, and redefine M in terms of the grid, thendraw the circles representing the area of the pedal triangle at each point ofthe grid. To define the grid, a system of axes is required. We shall take thedefault axes which are available for any diagram. To display them choose[attributes]Show axes. Next, activate the [attributes]Define Grid tool,and select the axes. A grid of points appears.

1

1

r = 0.35 cm

Area = 12.19 cm

AB

C

M

A'

C'

B' Figure 5.3 A grid is constructed,using the default axes for the diagram.M is then redefined as any point on

the grid.

M is still an independent, movable point in the plane; we shall redefineit so that it is limited to the grid points. Activate the [constructions]-Redefine object tool, then select M. Choose the option Point on an objectfrom the menu list that appears, and then select any point on the grid. M isnow constrained to the points of the grid.

The [constructions]Locus tool can now be used to construct the set ofcircles which are obtained by moving M around the grid. Select the circlethen the point M to obtain the locus of circles as M moves over the grid.

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It can be shown (see for example Geometry Revisited by H.M.S. Coxeterand S.L. Greitzer, Mathematical Association of America, section 1.9) thatthe contour lines of equal areas of the pedal triangles are circles with the

same centre as that of the circumcircle of ABC. In particular, triangleABC has zero area if M is on the circumcircle of ABC, or equivalently,points A, B and C are collinear if and only if M lies on the circumcircle ofABC.

r = 0.35 cm

Area = 12.19 cm

AB

C

M

A'

C'

B'

Figure 5.4 The distribution of the area of the pedal triangle as a function ofthe position of M.

Exercise 8 With M on the circumcircle of ABC, the three points A,B and C are collinear and ABC is called the Simson1 line for M (orWallace2 line this line was incorrectly attributed to Simson for many years,as it was in fact published in 1799 by Wallace). Construct the envelope ofSimson lines. (Use the [constructions]Locus tool.) This curve, which isinvariant under a rotation through 120, is called a deltoid (or tricuspoidor Steiners3 hypocycloid), since it has the form of the Greek letter . It is

1Robert Simson, 1687-17682William Wallace, 1768-18433Jakob Steiner, 1796-1863

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tangential to the three lines AB, BC and CA. It is an algebraic curve ofdegree 4.

Exercise* 9 For the deltoid of the previous exercise, construct the centre,the three points where the curve touches the three straight lines, and the

largest circle which can be inscribed in the curve.

A

B

C

M

Figure 5.5 The envelope of the Simson lines of triangle ABC is called adeltoid. It has the same symmetries as an equilateral triangle.

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Chapter 6

Functions

Graphs of functions are easy to construct in Cabri Geometry, thanks toits system of axes and expressions. The graph can then be used to study the

properties of the function. In this chapter, we shall study the polynomialfunction of degree 3, f(x) = x3 2x + 1

2.

First, display the coordinate axes, using [attributes]Show axes. Next,we need to create the corresponding expression on the drawing area.Once an expression has been placed on the drawing area, its value can becalculated for different values of its variables. For this function, activate[text and symbols]Expression, and type in x^3-2*x+1/2. The permittednames for variables are the letters: a,b,c...z.

Mark a point P, somewhere on the x-axis (using the [points]Point tool).

Display its coordinates by activating [measurement]Equation and Co-ordinates, then selecting P. The text displaying the coordinates is initiallyattached to P, and moves with the point. Using the [manipulation]-Pointer tool, the coordinates can be detached from P, and placed anywhereon the diagram. To return them to the point, click-and-drag close to P.

1

1

x^3-2*x+1/2

1

1

x^3-2*x+1/2

(3,49; 0,00)P

Figure 6.1 [Left]. The expression corresponding to the function is enteredon the diagram. [Right]. Point P is marked on the x-axis, and its coordinatesdisplayed using [measurement]Equation and Coordinates.


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Next, we need the value of f(x) when x is the x-coordinate of P. Activatethe [measurement]Apply an expression tool, and click on the expression,then the x-coordinate of P in the brackets. Here, the order is important.

1

1

x^3-2*x+1/2

(3,49; 0,00)

36,01

P

Figure 6.2 The [measurement]-Apply an expression tool is usedto calculate the value of f(x) at thex-coordinate of P.

This value is now transferred to the y-axis, using the [constructions]-Measurement transfer tool, and then selecting the value followed by they-axis. After this one merely has to construct the lines parallel to each of theaxes, through each of the marked points, using the [lines]Parallel line tool.Their point of intersection can be labelled M, and has coordinates (x, f(x)).In the following diagram we have moved P to a point (1.98, 0) so that M isvisible on the sheet. P can be moved during the construction of the lines.

Figure 6.3 Construction of thepoint M(x, f(x)) using measurementtransfer.

0.5

0.5

x^3-2*x+1/2

(1,89; 0,00)

3,50

P

M

The graph of the function is obtained as the locus of M as P movesalong the x-axis. It is constructed using the [constructions]Locus toolby selecting M then P. In order to see the interesting part of the function,the origin can be moved (using drag-and-drop), and the scale changed (bydragging-and-dropping any of the scale marks on the axis).

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0.5

0.5

x^3-2*x+1/2

(1,71; 0,00)

2,09

P

M

Figure 6.4 The graph of thefunction is finally created using the

tool [construction]Locus. The gridcan be moved and resized so that theinteresting part can be seen.

We shall now construct an approximation to the tangent to this curve at agiven point. For small values of h, it is known that

f(x) f(x + h) f(x h)2h

.

From the geometrical point of view, this approximation takes the gradientof the tangent to be the same as the gradient of the chord linking the pointson the curve whose x-coordinates are x h and x + h.

Using [text and symbols]Numerical Edit, a value for h is defined,0.3 here for example. This size of h enables the tangent to be constructedeasily. The value of h can then be changed to a smaller one, giving a betterapproximation to the tangent. Next, construct a point A on the x-axis,and the circle centre A, radius h. The circle is obtained by activating the[constructions]Compass tool then selecting the segment of length hfollowed by A. The two points of intersection of this circle with the x-axishave x-coordinates x h and x + h, if x is the x-coordinate of A. Drawthe three lines parallel to the y-axis ([constructions]Parallel line) whichpass through the two points of intersection, and the point A. The points ofintersection of these three lines with the curve provide the points B, B,

B+ which are points on the curve with x-coordinates x h, x, and x + h,respectively.

As the diagram is becoming rather cluttered, hide those elements whichare no longer being used. Activate the [attributes]Hide/show tool, andselect the elements to be hidden. Here, we should hide P, M, the twoconstruction lines for M, the coordinates of P, and the value of the functionat P. The hidden objects are displayed as marquees (marching antsoutlines), and are only visible when the [attributes]Hide/show tool isactive. In order to make a hidden object visible once more; just reselect itwhen this tool is active.

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0.5

0.5

x^3-2*x+1/2

h = 0,3

A

B-B

B+

0.5

0.5

x^3-2*x+1/2

h = 0,3

A

B

Figure 6.5 [Left]. The three points on the curveB, B , B+ with x-coordinates

x h, x, and x + h are constructed. [Right]. The approximation to the tangentat B, once the construction elements have been hidden.

The approximation to the tangent is now the line parallel to BB+ whichpasses through B. Construct the latter line using the [lines]Line tool, thenthe line parallel to it using [constructions]Parallel line. Now hide the linethrough BB+ and the other construction elements until only h, A, B, andthe tangent at B are visible.

It can be seen that the value h = 0.3 already gives a very good approx-imation to the tangent. Nevertheless, this can be improved by decreasing

the size of h, for example by taking 0.0001. If the values taken are toosmall, errors may well appear, due to the way that numbers are representedinternally in the computer, these being limited to about 16 significant figures(decimal system).

By moving the point A along the x-axis, it is possible to see the positionof the three roots of the equation f(x) = 0, the stationary points of f, andthe point of inflection of the curve.

For information, the three solutions of f(x) = 0 are approximatelyr1 = 1.52568, r2 = 0.25865, and r3 = 1.26703. The x-coordinates of thestationary points are e1 = 6/3 0.81649, and e2 = 6/3 0.81649.The point of inflection is at (0, 1/2).

Exercise 10 Using the gradient of the tangent, draw the graph whichapproximates to the curve of the gradient function.

Exercise* 11 The tangent cuts the x-axis at a point A with x-coordinatex, which is, in general, a better approximation to the root, provided A isalready in the neighbourhood of a root of f(x) = 0. This statement is thebasis of the iterative method known as the Newton1-Raphson2 method for

1Sir Isaac Newton, 1643-17272Joseph Raphson, 1648-1715

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finding the root of an equation. Construct the point A, then its iterateA by the same method, and compare the position of A to that of A. Inparticular, two positions can be found for A, other than the three roots,

for which A

and A coincide. For information, these are the two real rootsof a polynomial of degree 6, whose values are approximately 0.56293 and0.73727. It can also be seen that a poor choice of A can cause the methodto diverge, by choosing A so that A is one of the two points where thederivative is zero.

0.5

0.5

x^3-2*x+1/2

A

B

A'

B'

A''

Figure 6.6 The first two iterations of the Newton-Raphson method, startingfrom point A.

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Chapter 7

Tessellations I

We shall construct several tessellations of the plane, using polygons. Letus start with some simplified definitions, which are sufficient for the following

work. The reader who is interested can refer to the reference work Tilingsand Patterns by Branko Grunbaum and G.C. Shepherd, Freeman 1987. Alarge number of Internet sites also give information about tessellations andsymmetry groups.

We say that a set of closed plane shapes is a tessellation of the plane iftheir interior parts are non-overlapping, and the union of all the enclosedparts covers the entire plane. The parts of the plane used are called tiles.The intersection of two tiles which is a segment of a line or a curve is calledan edge, and the intersection of two or more tiles at a single point is called avertex.

For the tessellation P, we write S(P) for the set of isometries, f, of theplane such that the image of every tile of P under f is a tile of P. S(P) is agroup, called the symmetry group of the tessellation. There are several casesto be considered for such a group:

S(P) contains no translations. S(P) is then isomorphic to a cyclicgroup (possibly reduced to the identity element) generated by rotationthrough 2/n, or to a dihedral group, being the symmetry group of aregular polygon with n sides.

S(P) contains translations which are all collinear. S(P) is thenisomorphic to one of the seven frieze groups.

S(P) contains two vector translations which are non-collinear. ThenS(P) is isomorphic to one of the 17 wallpaper groups (or planecrystallographic groups), and the tessellation is said to be periodic.

If all the tiles of the tessellation can be obtained as isometries of a singletile, we say that the tessellation is monohedral.

In this part, we are only interested in the case of monohedral tessellationsby polygons.


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We shall first construct a monohedral tessellation of the plane by atriangle.

Construct a general triangle ABC, using the [lines]Triangle tool,then the midpoint, I, of one of its sides, BC for example, using the[constructions]Midpoint tool. Let D be the image of A under a half-turnabout I (point symmetry), which is created using the [transformations]-Symmetry tool, selecting first the object to be transformed: A, then thecentre: I.

AB

C

I

D

Figure 7.1 Starting with anytriangle ABC, its image is createdunder point symmetry about themidpoint of one of its sides (BChere). This produces a parallelogram

ABDC.

The quadrilateral ABDC is a parallelogram, and it can be used to

tessellate the plane. The two vectors AB and AC are created next, using the[lines]Vector tool, then these are used to duplicate the triangles ABC andBC D using the [transformations]Translation tool.

Figure 7.2 The[transformations]Translationtool is used to create the images of thetwo triangles under translation by thevectors

AB and

AC.

AB

C

The same approach can be used to tessellate the plane with any quadri-lateral, convex or otherwise. The image of the quadrilateral is created underrotation about the midpoint of one of its sides. This produces a hexagon

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whose sides are pairwise parallel, which is then used to tessellate the planeby translations.

Figure 7.3 The same type ofconstruction is used to tessellate theplane with any quadrilateral, convex orconcave, provided it is not a crossedquadrangle.

For other convex polygons, the situation is much more complex. It canbe shown that it is impossible to tessellate the plane with a convex polygonwith more than 6 sides. There are three types of convex hexagon whichwill tessellate the plane, and at least 14 types of convex pentagon, eachtype being defined by a set of constraints on the angles and the sides. Atthe present time, it is still not know if the 14 known types constitute thecomplete solution to the problem. The last of the 14 was discovered in 1985.As far as we know, the question of concave polygons has not been resolved.

Exercise 12 Construct a convex pentagon ABCDE, subject to thefollowing constraints: the angle at A is 60, at C it is 120, AB = AE, CB= CD. These constraints do not define a unique pentagon, but a family ofpentagons. For the construction there are at least three independent points.

Figure 7.4 Construction of apentagon under the constraints:A = 60, C = 120, AB = AE,and CB = BD. A, B, and C areindependent points in the plane.

A

B

E

D

C

Make successive rotations about A through an angle of 60 using the[transformations]Rotation tool. This tool requires selection of: the

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object to be transformed, an angle, and the centre of rotation, to constructa flower with 6 pentagonal petals. The angle required by the tool is anumber on the drawing area, which has previously been created using the

[text and symbols]Numerical Edit tool.

Figure 7.5 The basic pentagon isduplicated by rotation about centre A,through an angle of 60, to form a6-petalled flower.

These flowers can now be assembled, using translations, to tessellate theplane. This tessellation is type 5 according to the classification given inTilings and Patterns. It was first published by K. Reinhardt in 1918.

This tessellation is not only monohedral, that is to say that all the tilesare identical within an isometry, but it is also isohedral: all pentagons are

surrounded by the same pattern of pentagons in the tessellation.

Figure 7.6 The flowers areassembled by translations to cover theplane.

Exercise* 13 Construct a pentagon ABCDE with constraints: E = 90,A + D = 180, 2B D = 180, 2C + D = 360, EA = ED = AB + CD.Points A and E are independent points of the plane, and point I is free tomove on the arc of a circle.

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D

E A

I

B

C

Figure 7.7 A pentagon of type10, according to the classification in

Tilings and Patterns. This pentagon isthe basis for a monohedral tessellationof the plane.

The tessellation is constructed by first making three copies of the tile,using successive rotations through 90 about E, to obtain a truncatedsquare. These squares are then assembled in strips using translation in onedirection. The strips of squares are then separated by strips of pentagons, asshown below.

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Figure 7.8 A monohedral tessellation of the plane by convex pentagons. This

tessellation was created by Richard E. James III, following the publication ofan article by Martin Gardner in Scientific American in 1975. The completearticle can be found in Time travel and other mathematical bewilderments,Martin Gardner, Freeman 1987.

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Chapter 8

Tessellations II

This chapter makes reference to the definitions given in the chapterTessellations I.

There are sets of polygons which cannot be used to produce periodictessellations. The best known of these are the Penrose tessellations, namedafter the mathematician Roger Penrose1 who discovered them in 1974. Thesetiles are called Kites and Darts. A coloured motif is drawn on the tiles, andonly those assemblies that respect the correspondence of the colours areallowed. This eliminates all the periodic tessellations. These two tiles arequadrilaterals whose angle are multiples of = 36, and whose sides are oflength 1 and (phi), with = (1 +

5)/2. The colour motif shown here

was drawn by John Conway2, and gives astonishing curves that are invariantunder rotation through .

Figure 8.1 A Dart tile (left) and aKite tile (right).

36

216

36

72 72

72

72

144

The Kite and Dart tiles have lengthy constructions, so we shall createa construction macro which enables copies of these tiles to be depositedliberally across the drawing sheet by means of a single click.

A (construction) macro corresponds to a sub-section of a diagram. Itsdefinition is based on a set of initial objects, and a set of final objects whichare constructed uniquely from the initial objects. Once the macro has beendefined, it is available as a new tool in the [macros] toolbox. To completethe construction, the user has to select a set of objects, of the same type as

1Sir Roger Penrose, 1931-2John Horton Conway, 1937-


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the initial objects, when the tool reproduces the construction stored in themacro, using these selected objects.

When the macro is created, it is possible to name it, to design an icon forit, and to save it in a separate file. To use a macro created in one diagramin another, both diagrams are opened, and the macro is available to both.The macro is saved in a diagram file if it is either used or created in thatdiagram.

It is possible to overlay a macro definition, by defining a macro of thesame name, and constructing objects of the same type. When the userattempts to save the macro, Cabri Geometry asks whether it should replacethe previous macro, or supplement it. If the latter choice is made, either ofthe macros can be used. For example, a macro can be defined which startseither from two points or from a line segment.

We shall define a macro named Dart 1 L, which starts from two points,A and B, and constructs a Dart tile on AB, to the right of it when lookingat B from A, with AB as a short side of the tile (hence the 1 in the name)and an arc, one foot of which is further from A than from B (hence the L forlong). In similar fashion we shall define the macros Dart 1 S (short) whoseshape is the same as that of Dart 1 L, but where the foot of the arc is nearerto A than to B. Dart phi L, Dart phi S need to be constructed in the sameway, also the four corresponding macros for the Kite.

To define these macros, we must first construct the tiles, starting from

two points. So, let us take any two points, A and B, constructed with the[points]Point tool, which will be used to represent the side of length 1 inthe tiles. Next, construct the straight line AB, using [lines]Line, then theperpendicular to AB through A, using [constructions]Perpendicularline, and the circle through B, centre A using [curves]Circle and selectingfirst the centre A then the point on the circumference B. Finally, constructthe point of intersection C of the perpendicular to AB and the circle. Selectthe point of intersection above AB, using [points]Point.

We must now divide the circle into 10 equal sectors. Construct the pointB diametrically opposite to A, centre B, and A diametrically opposite to

B, centre A. (Use the [transformations]Symmetry tool; by first selectingthe point to be transformed, then the centre of symmetry.) If A is consideredas the origin on line AB, then B is at 1, B at 2, and A at 1. We alsoneed A, the midpoint of AA, using [constructions]Midpoint. A is at1/2 using the above conventions. Now construct the circle, centre A,which passes through C. This circle cuts the line AB in two points: P tothe left of A, and Q to the right of A. P and Q are on AB at and 1respectively. The perpendiculars to AB, through P and Q cut the circle,centre A, radius AB in four points: the vertices of a regular pentagon,whose fifth vertex is at B. Using symmetry, the vertices of the decagoncan be completed as shown below. From this, the angle of = 36 and the

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length = (1 +

5)/2 are constructed. These dimensions are intimatelylinked to the regular pentagon.

A B B'A' A''

C

P Q

Figure 8.2 Division of the circleinto 10 equal sectors.

Draw the circle, centre A, through P. The radius of this circle is then .Now use [attributes]Hide/Show to hide all those elements of constructionwhich do not appear in the diagram below. The vertices of the regulardecagon inscribed in the circle, radius , are labelled R, 1, 2, 3, 4, P, 6, 7, 8, 9.

Figure 8.3 The subdivision intoequal segments is transferred to thecircle radius , and the obsolete

elements of construction are hidden.

A BP Q R

1

23

4

6

7 8

9

The construction can be continued by taking note of the followingdiagram. The line segments linking P to points 2 and 8 are constructedusing [lines]Segment, then the two quadrilaterals are drawn with [lines]-Polygon. Next, the circles from which the arcs are to be selected are drawnusing [curves]Circle, and finally the arcs are drawn using [curves]Arc.An arc of a circle is defined by three points: one extremity, an intermediatepoint, and the second extremity. The points used to define the arcs cannow be hidden so that they do not appear when the macro is used. Theappearance of the arcs and the quadrilaterals can be changed, using[attributes]Thick, and [attributes]Colour.

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Using the earlier construction, we can define the macro Kite 1 L similarly.Using these two macros, we can start to construct the Sun tessellation,which has the same symmetry group as the regular pentagon.

Figure 8.6 The start of the Suntessellation, constructed using our twomacros.

Exercise 14 Define the other six macros listed in the text above andcontinue the Sun tessellation. Draw the Star tessellation, whose centre iscreated from five Dart tiles pointing towards the central vertex.

Exercise 15 List the 7 possible configurations of the Penrose tiles arounda vertex.

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PartThree

ReferenceSection


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Chapter 9

Objects andTools

This chapter lists the set of objects manipulated by Cabri Geometry, allthe different ways of creating them, and their attributes. The attributes and

how to change them is given in detail in the chapter Attributes.

All the objects can have an associated label. This consists of alphanumericcharacters attached to an object, for example, the name of a point. Whenan object is created, it can immediately be given a name, typed in on thekeyboard. The label can be changed subsequently, using the tool [text andsymbols]Label.

9.1 Point

The point object is the basis of all shapes. Cabri Geometry manipulatespoints in a Euclidean plane, with special treatment for points at infinity.

An independent, movable point can be marked on the plane, using the[points]Point tool and clicking on an empty part of the drawing area. Thepoint can then be moved anywhere in the plane, (using [manipulation]-Pointer).

A point can be created on a line (segment, line, ray...) or on a curve(circle, circular arc, conic, locus) either implicitly using the tool [points]-Point, or explicitly with [points]Point on an object. A point created this

way can be moved freely on the object.

Finally, one can create a point of intersection of two lines/curves, eitherimplicitly with [points]Point, or explicitly with [points]Intersectionpoints. In the latter case, all the points of intersection of the two objectsare constructed simultaneously.

The [constructions]Midpoint tool constructs the point which is midwaybetween two existing points, or the midpoint of a line segment or side of apolygon.

The [constructions]Measurement transfer tool transfers a length


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onto a ray (select the measurement and the ray), a vector (select themeasurement and the vector), an axis (select the measurement and the axis),a circle (select the measurement, the circle and a point on the circle), or a

polygon (select the measurement and the polygon). In all these cases a newpoint is constructed.

A point can be constructed as the image of a point under a transforma-tion, using one of the tools in the [transformations] toolbox. The objectpoint is selected first, then the elements that define the transformation.Inversion is the only tool in [transformations] which is not affine. In CabriGeometry inversion can only be applied to points.

When any other tool is used which requires the selection of a point, thiscan be done by selecting an existing point, by constructing a point implicitly(on a line or curve), or at the intersection of lines or curves. In this case, theoperation is the same as for the [points]Point tool.

When a line or ray is being created, the second point can be created onthe fly as an explicit point by holding down the Alt key until the positionfor the second point is selected.

The attributes of a point are its colour, shape, size, label, picture(optional).

9.2 Line

Cabri Geometry manipulates lines in the Euclidean plane, with the addi-tional possibility of a line of points at infinity if the treatment of infinity hasbeen activated.

The [lines]Line tool is used to create a line through a given point, byselecting first the point, then clicking anywhere else on the plane. The linecan then be made to pivot freely about the point. This tool can also be usedto construct a line through two points. The second point can be created onthe fly by holding down the Alt key.

In the case of a line defined by two points; if the two points coincide, theline is undefined.

The tools [constructions]Perpendicular line and [constructions]-Parallel line construct the unique perpendicular/parallel line in a direction(given by a segment, a line, a ray, a side of a polygon, a vector, or an axis),passing through a given point.

The [constructions]Perpendicular bisector tool creates the lineequidistant from two points, or the perpendicular bisector of a line segmentor side of a polygon.

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The [constructions]Angle bisector tool constructs the line whichbisects an angle. By selecting the three points A, B and C, the angle definedby the segments BA and BC is bisected. The second point selected must be

the vertex of the angle.

A line can be constructed as the image of another line under an affinetransformation by using the tools from the [transformations] toolbox.

The attributes of a line are its colour, thickness, line style, and label.

9.3 Line Segment

The [lines]Segment tool is used to construct the segment between two

points. If the two points coincide, the segment is still defined, but reducedto a point.

A line segment can be constructed of another line segment under an affinetransformation.

The attributes of a segment are its colour, thickness, line style, endpointstyle, label, and picture (optional).

9.4 Ray

The [lines]Ray tool is used to create a ray starting from a point. First thepoint is selected, then any free position of the sheet. The ray will then pivotfreely about the point. This tool will also construct a ray starting from afirst point and passing through a second point. Alternatively, the secondpoint can be created on the fly by holding down the Alt key.

In the case where a ray is defined by two points; if the two points coincide,the ray is undefined.

A ray can be constructed as the image of another ray under an affine

transformation.

The attributes of a ray are its colour, thickness, line style, and label.

9.5 Vector

A vector is represented by two endpoints. Consequently, a vector ismanipulated as if it is a line segment plus a direction (or sense) indicated byan arrow.

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The [lines]Vector tool uses two points to construct a vector. If the twopoints coincide, then the vector that has been defined is the zero vector.

The [constructions]Vector sum tool constructs a representation of theresultant of two vectors. The two vectors are selected, then the point wherethe resultant is to start.

A vector can be constructed as the image of another vector under an affinetransformation.

The attributes of a vector are its colour, thickness, line style, label, andpicture (optional).

9.6 Triangle

A triangle is a polygon with three vertices. Triangles and polygons aregenerated in the same way. Since the triangle is by far and away the mostfrequently used polygon, a special tool is available for them.

The [lines]Triangle tool uses three points to create a triangle. It ispossible to have triangles with zero area, or with two or three coincidentpoints.

An image triangle can be created of another triangle under an affinetransformation.

The attributes of a triangle are its colour, thickness, line style, fill colour,label, and picture (optional).

9.7 Polygon

In mathematics, the concept of a polygon can be defined in several ways. InCabri Geometry, we shall call a polygon the set of n segments

P1P2, P2P3 . . . P n1Pn, PnP1

defined by n points (n 3). Hence a Cabri-polygon is, above all, closed.The [lines]Polygon tool constructs a polygon using at least three points.

To finish off the construction, the first point created must be reselected,or the final point must be created by a double-click. If all the points arecollinear, it has zero area and is represented by a line segment.

The [lines]Regular polygon is used to construct regular polygons orstars. First select the centre of the polygon, then the first vertex. Thenumber of sides, and the interval between vertices for a star, can be chosen.

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In the final phase of the construction, a pop-up message follows the cursor todisplay the number of vertices, and the interval between consecutive vertices.For example, 5 indicates a regular pentagon, while 10/3 is a ten-branch

star, drawn by linking vertices 1, 4, 7, 10, 3, 6, 9, 2, 5, 8, and 1 of a regulardecagon.

A polygon can be constructed as the image of another polygon under anaffine transformation.

The attributes of a polygon are its colour, thickness, line style, fill colour,label, and its picture (optional) in the case of a quadrilateral.

9.8 Circle

The [curves]Circle tool creates a circle anywhere on the drawing area.First select the position of the centre, then a point somewhere else on thesheet. The radius of the circle can then be changed at will. Alternatively,the second point can be created on the fly by holding down the Alt key.

The [curves]Circle tool also constructs a circle by selecting first itscentre, then a point on the circumference.

A circle can be constructed as the image of another circle under an affinetransformation.

The attributes of a circle are its colour, thickness, line style, fill colour,and its label.

9.9 Circular arc

An arc of a circle is part of the circle which has two extremities, andcontains a third point. The [curves]Arc tool constructs an arc using threesuch points: the first is an extremity, the second is an internal point, andthe third is the other extremity. If the three points are collinear, the arc

becomes a line segment or the complement of a line segment (a line with agap in it), depending on the relative positions of the three points on the line.

An arc can be constructed as the image of another arc under an affinetransformation.

The attributes of an arc are its colour, thickness, line style, fill colour (ofthe associated segment), and label.

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9.10 Conic

Cabri Geometry enables all proper conics (ellipses, parabolas, hyperbolas) tobe manipulated in the Euclidean plane. Degenerate conics, consisting of twointersecting straight lines, are also possible.

The [curves]Conic tool constructs a conic through five points. If four ofthe points are collinear, or two of the points coincide, no conic is created. Incontrast, if only three points are collinear, two intersecting straight lines (adegenerate conic) are constructed.

A conic can be constructed as the image of another conic under an affinetransformation.

The attributes of a conic are its colour, thickness, line style, fill colour,and its label.

9.11 Locus

Different types of objects are created by Cabri Geometry under the namelocus. In general, a locus represents all those positions which can beassumed by an object A as a point M moves on an object. Normally, theconstruction of A makes use of the point M.

A locus is constructed using [constructions]Locus, first selecting theobject, A, then the variable point, M.

Object A can be one of the following types: point, line, ray, segment,vector, circle, arc, or conic. Point M can be a variable point on any type ofline or curve, including a locus, or even a free point on a grid.

Object A can equally well be a locus, when a set of loci is constructed. Inthis case, Cabri Geometrys performance can deteriorate significantly due tothe complexity of the diagram, and a warning message is displayed.

In the case where A is a line, ray, line segment, vector, or circle, thelocus is either the envelope of the lines, rays... or the entire set of theobjects, depending on whether or not the box Envelope has been tickedin the Preferences dialog box. (See the chapter on Preferences andCustomisation.) Vectors behave in just the same way as line segments forthe creation of a locus.

The envelope of a set of rays, line segments, or vectors is the same as theenvelope of the lines of which they are part, but restricted to those pointsthat they actually pass through.

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In the case where A is an arc or a conic, the locus is automatically the setof positions taken up by A.

The attributes of a locus are its colour, its thickness, its line style, itslabel, its construction method (envelope or set of positions) and its drawingmethod (continuous or set of points), the minimum number of positions tobe calculated.

9.12 Transformation

Cabri Geometry does not have an explicit object-type which is transform-ation. Rather, transformations are effected by the tools. Each tool, inapplying a transformation to an object requires various elements to define it(centre, axis, angle...). Cabri Geometry provides the usual affine and Euc-lidean transformations (enlargement, translation, reflection, point symmetry,rotation), as well as inversion.

In all cases, the object must be selected, and also the elements whichdefine the transformation. If the object to be transformed is of the sametype as one of the elements that define the transformation, it has to beselected first. In other cases, the order of selection is immaterial. Forexample, for the point symmetry transformation of point M with point Cas centre, M is selected first, then C. For point symmetry of line D withrespect to point C, selection can be in either order.

The object to undergo a transformation can be a point, any type of lineor curve, but not a locus. For inversion, only points can be selected. In thiscase, a locus can be used to create more complex images.

The tool [transformations]Reflection applies orthogonal symmetrywith respect to an axis. The object for transformation is selected and theline which is to be taken as the axis: line, ray, segment, vector, side ofpolygon, axis.

The tool [transformations]Symmetry applies symmetry with respectto a point (point symmetry or half-turn). The object for transformation isselected and the centre of symmetry (a point).

The tool [transformations]Translation applies a translation. Theobject is selected and the vector which defines the translation.

The tool [transformations]Enlargement applies an enlargement. Theobject to be enlarged is selected, the scale factor (a real number on thedrawing area), and the centre of enlargement (a point).

The tool [transformations]Rotation applies a rotation. The object to

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be rotated is selected, the angle of the rotation (a number on the drawingarea or an angle mark) and the centre of rotation (a point).

Finally, the tool [transformations]Inversion constructs the inverse of apoint with respect to a circle. The point for transformation is selected, thenthe circle which is invariant under the inversion.

9.13 Macro

A macro definition is based on a diagram. Once it has been defined, a macrocan be used in just the same way as any other tool, and reproduces theconstruction process using the initial elements that are selected by the user.

For example, a macro can be defined which will construct a square on agiven diagonal. To define the macro, first construct the square using anyline segment as the diagonal, then select the initial objects here the linesegment and the final objects here the square and finally save themacro. This is now a new tool which is in the [macros] toolbox, and whichrequires the selection of a segment, upon which it constructs the square. Theobjects which are created as part of the construction method are hidden,and cannot be displayed.

To define a macro, it can be seen that the corresponding construc-tion must already exist. With the construction displayed on-screen, the

[macros]Initial objects tool is activated and the initial objects of theconstruction are selected. For objects of the same type, the order of selectionis important, and the same order will be required when the macro is used.For initial objects of different types, the order of selection is immaterial. Theset of initial objects in the diagram flash or are displayed with marchingants outlines. To add objects to or remove them from the list of initialobjects, simply click on them.

When the selection of initial objects has been completed, the final objectsmust be defined. The [macros]Final objects tool is used, with selectionof members of the final objects set being made in the same way as before.

Until the macro is saved, the sets of initial and final objects are held inmemory and can be changed at will.

Finally, the macro just has to be defined, using [macros]Define Macro.Cabri Geometry first checks that the final objects can indeed be constructedfrom the set of initial objects selected. If this is not the case, the macro willnot be defined, and an error message is displayed: This macro-constructionis not consistent. Cabri cannot determine all final objects with the

given initial objects.

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If the macro is a consistent entity, a dialog box is displayed, for the user toedit the attributes of the macro. The only item which must be completed isthe name of the construction. All other attributes are optional.

Name of the construction. The name of the macro as it will appearin the [macros] toolbox.

Name of first final object. The name appears to identify the object,as the cursor is moved over the drawing area. For example if themacro constructs the perpendicular bisector of the segment joiningtwo points, the name of the final object could be This perpendicularbisector.

Password. If a password is allocated to the macro, its intermediateconstruction objects are inaccessible from the History window whichdisplays the macro in text form. (This window is opened with the F10key.)

An icon for the construction can be created in the other part of the dialogbox. Clicking on the Save button enables the macro to be saved in astand-alone file. The macro is saved both in the diagram where it has beencreated, and in any diagram where it is used. A macro which is loaded intoa diagram is available to all other diagrams that are open simultaneously.

If a macro has the same name, and constructs the same type of finalobjects as one already defined, Cabri Geometry gives the user the choiceof adding the new one to that already existing, or replacing it. If the user

decides to add it to the one already existing, Cabri Geometry will choosethe appropriate macro to use according to the initial objects selected. Forexample, if a macro is defined with two points as initial objects, anothermacro could be added to it which is identical apart from having a segmentas the initial object. The standard tools [constructions]Perpendicularbisector and [constructions]Midpoint... have been added to in this way.

To use the macro, the corresponding tool in the [macros] toolbox isactivated, then the initial objects are selected. When the initial objectshave been selected, the construction follows automatically, and the newset of final objects appears. The objects which are created as part of the

construction method are hidden, and cannot be displayed by using the[attributes]Hide/Show tool.

When a macro is used, an object can be defined as an implicit argument ofthe macro by holding down the Alt key when the object is selected. Whenthe macro is used in future, it is no longer necessary to select this object asan argument: it will be selected automatically.

If, for example, a macro requires the selection of two points and a circle,and on one occasion two points are selected, and the Alt key is held downwhile a circle is selected, then in future the macro will only require the

selection of two points, the circle being selected automatically.

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If the attributes of final objects are not the same as the default attributeswhen the macro is defined, they will be saved with the macro and applied tothose objects which are created when the macro is used.

9.14 Number

A number can be any real number, displayed on the drawing area, and canhave an associated unit. Numbers are displayed as dynamic elements withintext messages. (See section 9.17, Text.) When a number is created, CabriGeometry creates a text message whose sole content is the number. The textmessage can be edited subsequently.

The [text and symbols]Numerical Edit tool enables the user to enterthe number directly onto the drawing area. The number can then be editedand animated. The up and down arrows to the right of the number, andalso animation can be used to modify its value, changing the digit to theleft of the cursor position in the edit window. For example, if the number is30.29 and the cursor is between the 2 and the 9, animation or the use of thearrows will change the value of the number by steps of 0.1.

The [measurement]Distance and length tool creates a numberrepresenting the distance between: two points, a point and a line, a pointand a circle; the length of a segment, a vector, an arc of a circle; theperimeter of a polygon; or the circumference of a circle or ellipse. The

resulting value is given in cm as the default unit of measurement.

The [measurement]Area tool creates a number to represent the areaof a polygon, circle or ellipse. The text message includes a unit of area, thedefault unit being cm2.

The [measurement]Slope tool measures the gradient of a line, ray, linesegment or vector. The value is dimensionless.

The [measurement]Angle tool measures the size of an angle. Thearguments required are three points: A, O and B in that order, where thesides containing the angle are OA and OB, or a single argument, being thealready existing mark of an angle.

The [measurement]Calculate tool is used to perform calculations onnumbers displayed in the drawing area, the constants pi and infinity, ordirectly entered real values. The usual operators can be used: x + y, x y,x y, x/y, x, xy, (x). The calculator also recognises the following standardfunctions: abs(x), sqrt(x), sin(x), cos(x), tan(x), arcsin(x), arccos(x),arctan(x), sinh(x), cosh(x), tanh(x), arcsinh(x), arccosh(x), arctanh(x),ln(x), log(x), exp(x), min(x, y), max(x, y), ceil(x), floor(x), round(x),sign(x), random(x, y). Some variants of these spellings are recognised: aninitial capital letter, asin, sh, ash, argsh...

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The inverses of functions can be used by combining the inv button withthe function button. For example, to use the arcsin function, click on thebuttons inv then sin. This extends to inv-sqrt which gives sqr, inv-ln

which gives exp (ex

) and inv-log which gives 10x

.

Apart from the standard operators, for which the syntax is well-known,floor(x) returns the largest integer less than or equal to x, ceil(x) returnsthe smallest integer greater than or equal to x, round(x) returns the integernearest to x whos

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