By the Editors: Doris Schattschneider and James King

Mathematicians all know the power that a figure can provide--often a quick
sketch or a diagram can make everything clear. We say "I see" and mean "I see
and understand." In geometry, figures seem to be essential for most
descriptions and proofs. Yet mathematicians also know the danger in relying on
figures--inevitably, extra assumptions are made (suggested by a sketch),
special cases are missed (omitted from a sketch), or absurd results are derived
(from an inaccurate sketch). A classic case of the last occurence is the
following oft-cited "theorem" and proof
^{2}.

**Theorem. ***All triangles are isosceles*.

*Proof:* Let *ABC* be a triangle with *l* the angle bisector of
*A*, *m* the perpendicular bisector of *BC* cutting *BC* at
midpoint *E*, and *D* the intersection of *l *and *m*.
From *D*, draw perpendiculars to *AB* and *AC*, cutting them at
*F* and *G*, respectively. Finally, draw *DB* and *DC*.
Figure 1 shows a sketch of the situation.

Triangle *ADF* is congruent to Triangle *ADG *(aas), so *AF*
is congruent to *AG* and *DF *is congruent to * DG.*
Triangle *BDE* is congruent to *Triangle CDE* (sas), so *BD*
is congruent to *CD*. This implies Triangle *BDF* is congruent to
Triangle *CDG* (hyp-leg), so *FB* is congruent to *GC*. Thus
*AB* is congruent to *AF* + *FB* = *AG* + *GC*
is congruent to *AC*, and so Triangle *ABC* is isosceles. QED

What is wrong with this proof? Absolutely nothing is wrong with the chain of
reasoning--the conclusion must follow if the figure is accurate. But even
though the sketch is plausible, the configuration cannot occur as drawn.
*D*, the intersection of *l* and *m*, is never inside
Triangle *ABC*.

This fallacious proof has been offered by some as a reason to ban the drawing
of *any *figures! But they miss the point.

Quite to the contrary, this example offers compelling reasons to emphasize the
need for acurately constructed figures, rather than quick sketches, especially
when they are to be used in deductive aguments. As E. P. Northrop put it, "It
shows how easily a logical argument can be swayed by what the eye sees in the
figure and so emphasizes the importance of drawing a figure correctly, noting
with care the relative positions of points esential to the proof. Had we at
the start actually constructed--by means of rule and compass--the angle
bisector and the various perpendiculars, we should have saved ourselves a good
deal of trouble."^{3}
The "proof" also underscores the need for a
variety of constructions, rather than just one. (Authors who discuss this
particular "proof" display a variety of inaccurate sketches of various cases as
well as accurate ones obtained by construction.)

Even careful constructions can be inaccurate, as anyone who has labored with a wobbly compass or a thick lead can testify. Complicated constructions are time-consuming and difficult to make precise. The time (and tedium) required to make a variety of figures by hand to illustrate various cases of a geometric configuration is beyond what almost anyone is willing to spend.

With dynamic geometry software, the proof above would be most unlikely. With
the software, an investigator could make a single accurate sketch of the
geometric configuration described in which the constraints on *l* (angle
bisector of *A*), *m* (perpendicular bisector of *BC*), *D*
(intersection of *l* and *m*), *F* and *G* (feet of
perpendiculars from* D* to *AB* and *AC*, respectively) are
specified. Then the shape of Triangle *ABC* could be deformed at will
with all the constraints being preserved in the newly created configuration.
Almost instantaneously, a (seemingly infinite) family of accurately drawn
configurations would be produced by manipulating a vertex of the elastic
figure. The investigator would see that the only time that *l* and
*m* do not intersect outside Triangle *ABC* is when *l* and
*m* coincide (and *B = F*, *C = G*), and that is exactly the
case in which Triangle *ABC* is isosceles. Moreover, in all but that
special case, the equation in the last line of the "proof" is false. Figure 2
captures only a few of the different configurations that occur.

Today there are new generations of personal computers in the schools (and on
teachers' and researchers' desktops and laps). The dynamic geometry software
that has been developed takes advantage of the new hardware (with vastly
increased memory, speed, and mouse interface for graphics) and builds on the
early ideas. The most commonly used dynamic geometry software is *The
Geometer's Sketchpad*, *Cabri Geometry II*, *Geometry Inventor*,
and the successor to the Supposers, the *Geometric SuperSupposer*. These
all provide means for accurate construction (and measurement) of geometric
configurations (of points, segments, rays, lines, arcs, circles) and have the
ability to replay a construction. What is most exciting about the new
software, however, is that unconstrained parts of the configurations (arbitrary
segments or points, for example, that are not dependent on any other objects)
are moveable--they can literally be grabbed with a cursor (using the mouse) and
be dragged or stretched--and as they move, all other objects in the
configuration automatically self-adjust, preserving all dependent
relationships and constraints. In doing this, a smooth sequence of instances
of the configuration can be viewed, morphing contnuously from the first drawing
to the final one that remains when the dragging is stopped. If the
configuration is a general one (beginning with an "arbitrary" line, circle, or
polygon, for instance) there can be no question that there are many (even an
infinite number) of cases, and the visual evidence of invariant properties
(such as concurrence of lines, collinearity of points, or ratios of certain
measurements) is compelling.

In addition to construction capabilities, the programs allow certain affine transformations (translations, reflections, rotations, dilations) to act on figures or their parts. These transformations can be determined by "fixed" data or by "dynamic" data, such as a rotation by a fixed angle (e.g. 45 degrees) or by a moveable angle that is in some sketched triangle, for instance. The software allows loci to be traced as parts of figures are dragged freely or guided along a specified circle or segment. The programs also have the special feature of being able to "record" the sequence of steps that leads to a particular configuration and then be able to use that construction as a "macro" tool in producing new configurations. For example, a script or macro that produced a tangent to a circle from a point outside the circle might be employed several times in producing a complicated figure. The software has components that link synthetic geometric constructions to analytic equations, coordinate representations, and graphs. Conics are also implemented in at least one program. Information on all of the software discussed by our authors is at the back of this volume.

Although this volume is printed in a conventional manner, with illustrations in
every article, most of the illustrations beg to be played with. You will read
descriptions of how certain configurations behave when manipulated, but not be
able to tweak the diagram on the printed page. We want you to be able to
experience some of the explorations described by our authors. To make that
possible, dynamic sketches that use *Geometer's Sketchpad* or *Cabri
II* have been made available by several of the authors and are posted on this
Web site maintained by the
Mathematics Forum.

This page can also be accessed by visiting the MAA web site.

For those who do not have access to the software, demonstration copies of the software can be downloaded and used to open the sketches.

**Overview of Contents**

Dynamic geometry software has had a profound effect on classroom teaching wherever it has been introduced. Although not originally intended by its developers, it has also become an indispensible research tool for mathematicians and scientists. The papers in this volume give a good idea of the ways in which the software can be used, and some of the effects it can have. Authors also make clear that the software raises various questions for teaching and research, and its continuing evolution raises questions on the design of the software itself. Many of the papers here expand on presentations that were given at the annual meeting of the MAA in San Francisco in January 1995 in a special session devoted to dynamic geometry.

In an attempt to give an overview of the contents of the papers, we address the basic question "What is dynamic geometry software good for?"

**Accuracy of construction**. Dynamic geometry software provides an
*accurate constructor* for any ruler-and-straightedge construction in
Euclidean geometry, any configuration that can arise by applying affine
transformations (isometries and dilations) to a Euclidean construction, or any
locus of an object (or set of objects) that arises when part of a construction
is moved along some path. Reliance on the accuracy of the software's geometric
sketches and measurements is so basic that it is not discussed explicitly in
any article, but it is implicit in every discussion. (Accuracy is, of course,
limited to the tolerences of internal computation, screen display, allowed
numerical display, and printer fidelity. And accuracy is occasionally
diminished by the choice of heuristic for an algorithm or by pesky bugs that
are bound to occur in such complex software.) Arnold and Bernadette Perham
offer several suggestions for investigations of constructions that exhibit the
collinearity of 3 points or exhibit invariance of the ratio of two segments.
Tony Hampson describes how the software easily implements the techniques for
perspective drawing and dramatically displays the principle of parallel lines
converging at vanishing points, even as the viewpoint is changed. In every
article, the sketches are produced by software; these attest to the author's
reliance on it as a faithful constructor.

**Visualization**. As a demonstration tool in the classroom, dynamic
geometry software can help students *see* what is meant by a general fact.
Douglas Brumbaugh tells how an animated triangle got the attention of middle
school students and demonstrated that the area of a triangle is determined by
its base and height. Jim Morrow observes, "Dynamic geometry software can be
used to aid the process of visualization in all mathematics classes, not just
in the study of geometry. Students can construct, revise, and continuously
vary geometric sketches. While visualization in itself is a powerful
problem-solving tool, the capacity for students to make instantaneous and
precise variations to their own visual representations adds a new dynamic
dimension whose implications are only beginning to be understood." His
several examples illustrate how the abstract concepts of variable, function,
and invariance take on meaning through visualization. Al Cuoco and Paul
Goldenberg observe "By allowing students to investigate continuous variation
directly (without intermediary algebraic calculation), dynamic geometry
environments can be used to help students build mental constructs that are
useful (even prerequisite) skills for analytic thinking." Cathy Gorini shows
how the software can help students in calculus see (and vary continuously)
relationships between quantities in min-max applications and related-rates
problems.

**Exploration and Discovery**. In a traditional geometry course, students
are told definitions and theorems and assigned problems and proofs; they do
not experience the discovery of geometric relationships, nor invent any
mathematics. As Schwartz and Yerushalmy observed, "This constitutes a kind of
satire on the nature of mathematical thinking and the way new mathematics is
made."^{6}
Dynamic geometry software is perfectly suited for
exploration and discovery--either guided, or completely open-ended (a.k.a.
research). Tim Garry observes that it allows students to "test their own
mathematical ideas and conjectures in a visual, efficient and dynamic manner
and--in the process--be more fully engaged in their own learning." The range
of investigation and the amount of guidance provided varies greatly with the
level and experience of the students (or researchers).

Our authors provide a variety of of examples of investigations for students and in many cases, discuss the process and the outcomes of the tasks. For example, Tim Garry had his students investigate the ratio of perimeter to diameter for regular polygons and also explore questions concerning circles tangent to a given circle. Zhonghong Jiang and Edwin McClintock had preservice teachers find the shortest path (with restrictions) that connects towns A and B separated by one or more rivers and also find various constant ratios that arise in a construction of a triangle within a triangle. Students using the software often discover surprising things that are not in any text, and not known to the teacher. As Kathy Boehm points out, "Any teacher using...dynamic geometry software needs to be prepared for students to ask unexpected questions. I have had to answer, 'I don't know; let me get back to you,' many times." Michael Keyton describes the process of his students moving from guided discovery of well-known properties of families of quadrilaterals to open-ended investigation of new families of quadrilaterals (with newly-invented names) proposed by the students themselves. Fadia Harik gives her students (often teachers) tasks with deliberately ambigious directions and many possible solutions, designed to prompt questions and guide the process of formulating hypotheses.

Although it may seem a cliché, dynamic geometry software
*empowers*. Students can get hooked pursuing open-ended problems.
Michael Keyton reports that some of his students have come back after
graduating to tell him some new result on an unfinished investigation begun in
his class. Although ninth-grade student Ryan Morgan's discovery using dynamic
geometry software received national
attention^{7},
there are many more
students who have experienced a similar thrill of discovery. And not only
students. Douglas Hofstadter gives a riveting account here of his own pursuit
of a geometric problem that tantalized him. Mike de Villiers describes how,
with dynamic geometry software, he could follow the irresistible lure of "what
if?" to find some new results related to known theorems.

**Proof**. While dynamic geometry software cannot actually produce proofs,
the experimental evidence it provides produces strong conviction which can
motivate the desire for proof. Dan Bennett was tantalized by a 40-year-old
unsolved problem from the *American Mathematical Monthly* for which there
was compelling computer evidence of its truth; he was determined to find a
proof (and succeeded). Mike de Villiers discusses the synergy between using
dynamic geometry software and proving the conjectures that spring from its use.
Conviction is necessary for undertaking the (often difficult) search for a
proof, he contends, and in addition, the software may even give insight into
geometric behavior that can help with a proof. Subtle geometric relationships
may be not at all obvious, but be revealed in experimenting with dynamic
figures. Some teachers have been reluctant to use the software because they
fear that visually convincing evidence will replace proofs of theorems. Our
authors describe the opposite situation: when students make their own
conjectures based on their explorations with the software, they know it is not
enough to stop with the evidence; they need to devise a proof. Zhonghong Jiang
and Edwin McClintock record for us the process by which their students arrived
at proofs of their conjectures.

**Transformations**. Dynamic geometry software can *transform* figures
in front of your eyes. Isometries and similarities are important examples of
functions. In witnessing the action of these transformations moving and
scaling figures, students see that functions are not synonymous with symbolic
formulas. Doris Schattschneider points out that students can visually test
fundamental properties of composition such as commutativity and inverses, as
well as other abstract concepts encountered in the theory of groups. Jim Parks
describes how students can identify a transformation by producing a sequence of
points in its orbit.

Ross Finney's 1970 article "Dynamic Proofs of
Euclidean Theorems"^{8}
(written long before the current dynamic geometry software was even an idea)
begins, "Simple observations about [affine] transformations of the plane lead
to elegant proofs of unusual Euclidean theorems." Proofs such as Finney's can
be brought to life with the software--to show that one figure is the
transformed image of another, one only needs to perform the transformation to
see if the two figures coincide. Jim King's article is devoted to proofs that
rely on similarities. Mike de Villiers also gives proofs that depend on the
action of transformations.

**Loci**. It is virtually impossible for most people to imagine a point
moving in a configuration (in which other several parts may also may be moving)
and be able to describe the locus of the point's path as it travels. Dynamic
geometry software, with its built-in feature to trace the locus of any
specified object is ideally suited to show how a locus is generated and to
reveal the shape of its traced path. Except for the most simple loci (circles,
as the locus of points equidistant from a fixed center, and perhaps the
conics), this rich subject has been avoided in most geometry texts. In fact,
most of the classic curves arose as loci, and one must go to long out-of-print
books to find these nonanalytic descriptions of them. Classic locus problems
and intriguing generalizations of these, as well as surprising new loci, are
discussed by our authors. Hampson's students investigate conics. Cuoco and
Goldenberg describe various locus experiments--they create generalized
ellipses, explore a locomotion problem, and solve some optimization problems by
means of loci. Heinz Schumann and David Green describe some classic and
non-classic curves that are constructed as loci and also provides some useful
applications of loci. John Olive describes how the Joukowski transformation
(which uses inversion in a circle) produces airfoil shapes as loci.

**Simulation**. Dynamic geometry software's special features of dragging,
animation (of points on line segments or on circles), tracing loci, and random
point generation provide many opportunities to simulate a surprising variety of
situations. Our authors present imaginative examples: moving robot arms (Jim
Morrow), a sine tracer (Tim Garry), a 2-dimensional random walk machine (Tim
Garry), and an airfoil tracer (John Olive). David Dennis and Jere Confrey
simulate a mechanical linkage devised by Rene Descartes (1637) that produces
points in a geometric series and then pair those points with points in an
arithmetic series to produce logarithmic curves. In this simulation, the
curves can "flex and bend as the arithmetic and geometric sequences are
manipulated," since the manipulation effectively changes the base of the
logarithm. Ben Backus describes some of the special simulations devised to
help in the teaching of optics and in vision research. Susan Addington and
Stuart Levy use a 3D interactive viewer, *Geomview*,* * to produce a
simulation of what is seen in an Ames room, a distorted room-size box that
tricks our perception. Fadia Harik uses *The Physics Explorer: One Body*
to simulate the action of a billiard ball bouncing off the sides of a square
table.

**Microworlds**. Dynamic geometry software produces an environment in which
Euclidean geometry can be explored. Three papers in our collection, all by
designers of this type of software, discuss new environments that can be
created with software. Jean-Marie Laborde
(*Cabri-géomètre*) and Nick Jackiw (*The Geometer's
Sketchpad*) each discuss ways in which their differing software can create a
"Poincaré world" of hyperbolic geometry. Jackiw also discusses how
other microworlds can be created through the use of scripts which produce new
"tools" that replace the Euclidean tools and allow exploration fully within a
new geometry. Allen and Trilling describe a program
(*GéoSpécif*) under development that adds a new dimension to
the dynamic geometry programs currently in use. Their program takes as input
the specifications for a construction and then automatically produces the
construction (if it is possible); using this, students can explore to discover
various dependent relationships among the parts of the construction. All these
software developers acknowledge the challenges that dynamic geometry software
presents--there is a delicate balance between what might be desired and what is
feasible, and there is an ongoing dialogue as to what should be the design of
software that can best enhance learning for all.

**Notes and References**

1. Here, and throughout this volume, the term "dynamic geometry" is used in its broadest sense. Although Key Curriculum Press has trademarked the phrase, our authors' use of the phrase does not refer to any particular computer software or published software support materials.

2.
This fallacious proof has been in the literature for over 100 years; it is
probably even older. It appears, with many "cases" discussed, in the following
sources: *Mathematical Recreations and Essays*, by W.W. Rouse Ball and
H.S.M. Coxeter, 13^{th} ed., Dover 1987 (1^{st} edition 1892);
*Riddles in Mathematics*, by E.P. Northrop, Van Nostrand, 1944;
*Fallacies in Mathematics*, by E.A. Maxwell, Cambridge [Eng.] University
Press, 1959; and *Mistakes in Geometric Proofs*, by IA.S. Dubnov, Heath,
1963.

3. Northrop (see 1. above), page 101.

4.
J. Schwartz and M. Yerushalmy, designers, *The Geometric Supposers*.
Pleasantville, NY: Sunburst Communications, 1983-1991.

5.
D. Chazen and R. Houde, *How to Use Conjecturing and Microcomputers to
Teach Geometry*, National Council of Teachers of Mathematics, 1989.

6.
J.L. Schwartz and M. Yerushalmy, "Using Microcomputers to Restore Invention
to the Learning of Mathematics," *Contributors to Thinking*, D. Perkins
and R. Nickerson, eds. Lawrence Erlbaum Associates, Hillsdale, NJ, 1986, pp.
293-298.

7.
T. Watanabe, R. Hanson, and F. D. Nowosielski, "Morgan's Theorem,"
*Mathematics Teacher*, 89 (May 1996) 420-423.

8.
R. Finney, "Dynamic Proofs of Euclidean Theorems," *Mathematics
Magazine* 43 (1970) 177-185.

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