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Re: Request: examples of hard to prove euclidian geometry
Posted:
Jul 7, 2001 10:01 PM
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Humberto:
You may want to consult Euclid in the Dover paperback
edition of, "The Thirteen Books of Euclid." The unabridged
works of Euclid are presented in two volumes, including all the
axioms, postulates, definitions, propositions, lemmas, and, also,
offerings by other geometers. The editing, commentaries, and
histories of the concepts, by the editor, Sir Thomas Heath,
are superb.
Proofs written in word concepts are often shorter than proofs
written as mathematical descriptions. That is because the
concepts used in geometry are more universal than the
more particular concepts of mathematics. Proofs written using
mathematics must also use the fundamental concepts of
geometry. After the proof is written mathematics can provide
useful examples, however. The verbal proofs, terms, and
expressly Aristotelian logical structures that are used by
Euclid function at a conceptual level that is more fundamental
than mathematics, and upon which mathematics in a large
part is based.
Ralph Hertle
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Humberto Jose Bortolossi wrote:
> Dear friends,
>
> I would like to collect some hard euclidian geometry theorems (problems)
> so it would be nice if you could contribute with your opinion: please
> give examples of hard euclidian geometry (theorems) problems.
>
> Well, of course, what is hard for a person, it may not be hard for
> another person.
> To measure ``hardness'': some computer geometry provers convert geometry
> problems
> in multivariate complex polynomials and, if we may prove that such
> polynomial vanishes, we have proved the theorem. The degree of the
> polynomial may vary and this would be a way to measure how difficulty a
> geometry problem is.
>
> For example, the Napoleon theorem is expressed as a complex theorem of
> degree one!
>
> Any examples, suggestions or references are welcome!
>
> Thanks in advance, Humberto.
> hjbortol@mat.puc-rio.br
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