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Re: proof in geometry
Posted:
Apr 26, 2000 9:06 AM


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 2column 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 halfturn symmetry.
IF you have a quad (nonself intersetion) with opposite sides equal THEN the congruence of the two halves (created by a diagonal) IS the halfturn.
Given any other set of appropriate information on a quadrilateral (e.g. opposite angles equal) THEN I prove there is a halfturn 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 ÃÂforms of thoughtÃÂ 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 > >



