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Topic: proof in geometry
Replies: 6   Last Post: May 1, 2000 6:53 PM

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Walter Whiteley

Posts: 146
Registered: 12/3/04
Re: proof in geometry
Posted: Apr 26, 2000 9:06 AM
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I have been thinking about some of these issues. Won't call
it formal research - but here is one of my reflections (pun intended).

I am a geometer, with a view that the core of geometry lies in
the transformations (Klein's definition and hierarchy).

I have observed that the thinking that I do within the dynamic
geometry context (and the visual context more generally)
observes transformations, symmetries, etc. which change the
configuration to another configuration.

Often I observe 'proof problems' in texts which cry out for
use of transformations, but are placed inside a set of
restrictions where the accepted reasons (as in a 2-column proof)
are reasons which do not use the symmetries. They are logical
exercises with a very limited (intentionally) use of diagrams,
the visual information etc. Find triangles with three
appropriate measurements in common - cite a congruence theorem.
[Ps. do NOT look for what transformation the congruence
is talking about - that will be useless to you.]
Find four points on a circle, observe that angles subtended by
the same chord are the same. Find two triangles with two common
angles, note that the third angle is the same.

SOMETIMES, we go a bit further, and use the diagram as an excuse
to write down some algebra and actually practice algebra.

When one pays closer attention to the visual route, and the
visual reasoning, the techniques (and sometimes the questions)
shift.

For example, to ME the essential definition of a parallelogram
is a quadrilateral with a half-turn symmetry.

IF you have a quad (non-self intersetion) with opposite sides
equal THEN the congruence of the two halves (created by a diagonal)
IS the half-turn.

Given any other set of appropriate information on a quadrilateral
(e.g. opposite angles equal) THEN I prove there is a half-turn symmetry.

THEN I can conclude any of the relevant other equalities FROM THE SYMMETRY.

Similarly, when working with isoceles triangles (or similar
information), I typically
prove that there is a reflective symmetry about the right bisector of
the bottom (or sometimes the angle bisector of the top, as relevant)
and then conclude all the other relevant properties from the symmetry.

Some people may object that this is not what Euclid was doing.
I would reply that:
(a) it is appropriate to the instruments we now have
(as Euclid's were appropriate to his instruments);
(b) it is an initiation in how actual geometers work on actual
geometry problems these days;
(c) it builds the visual and visual reasoning skills which
people need, and which historically many cultures have had.

Note that I AM a research geometer. I am also interested
in diagrammatic / visual reasoning. (MY PhD was on
Logic and Invariant Theory - essentially some logical
questions on the foundations of analytic geometry). One of the reasons
people have distrusted visual reasoning is the types of reasoning
which were being studied. It is my observation that symmetry,
transformation arguments are easier to give correctly, via
visual reasoning, that some of the more traditional geometry
exercises.

Working with visual proofs and explanations, I have:
(a) changed the way I see;
(b) changed the questions I ask;
(c) changed the answers I give, even to some traditional questions;
(d) changed the processes I use to move from questions to answers.

Does this respond at all to YOUR questions?

Walter Whiteley
York University
Toronto, Ontario

>
> I have being working with dynamical geometry in a Euclidian Geometry
> course. The environment works very well for conjecturing. But for
> proving the results that are visually stable... that is another PROBLEM.
>
> So I am interested in research results that focus on the question: how
> to promote in such environment “forms of thought” which could empower
> students in creating proofs. If too general , maybe some key words would
> be: theorems in motion, recasting of classical theorems, reasoning
> beyond indutive and deductive...
> Thanks for your attention
> Maria Alice
>
>






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