Subject: RE: Pythagorean Theorem
From: Stefan Baratto (sbaratto@earthlink.net)
Date: Wed Mar 22 2000 - 13:38:47 EST
I was actually looking for something more geometric in nature to share with
a class. I think what struck me as oddest was that I hadn't seen one and
could only come up with proofs that required the "only if" portion of the
proof as a given.
On Wednesday, March 22, 2000 1:11 PM, RWW Taylor
[SMTP:RWTNTS@ritvax.isc.rit.edu] wrote:
> Stefan Baratto writes:
>
> > I am trying to find a proof of the converse of the Pythagorean Theorem
that
> > doesn't require the Pythagorean Theorem as a given. That is, if a
triangle
> > has sides whose lengths satisfy the relationship a^2 + b^2 = c^2 then
it is
> > necessarily a right triangle. The only proofs I could find (or deduce)
> > require one to assume that if a triangle is right, then the sum of the
> > squares of the lengths of the legs is equal to the square of the length
of
> > the hypotenuse.
>
> The question is what degree of "purity" is desired here? Classically,
there
> was no mention of "squaring lengths", and squares were always thought of
as
> _areas_, really a different _data type_ (to use a contemporary term).
Euclid
> always talked about the "square on the hypotenuse, etc." and the
conception was
> an actual, constructed square. Figures could be compared by assuming
certain
> principles of area measure, but there was no necessary connection between
> linear measure and area measure. A proof of the converse, as desired,
that
> sticks to this careful understanding would be the highest level of
response,
> but would likely need to be very subtle.
>
> Right down to the middle ages, powers of quantities were conceived of as
having
> geometrical significance, and different powers could not logically be
combined
> by addition. Today, of course, all is number and we have no trouble
forming
> arbitrary polynomials at will. I expect that a proof of the desired
theorem
> would not be difficult to come up with in a coordinate geometry setting
> (though I don't have one in mind just at the moment). Would such a proof
be
> satisfactory?
>
> RWW Taylor
> National Technical Institute for the Deaf
> Rochester Institute of Technology
> Rochester NY 14623
>
> >>>> The plural of mongoose begins with p. <<<<
>
>
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