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RE: Generalization of the Pyphagorean Theorem?
Posted:
Jan 4, 2005 3:40 AM
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> -----Original Message-----
> From: owner-geometry-puzzles@mathforum.org [<a href="mailto://owner-geometry-">mailto://owner-geometry-</a>
> puzzles@mathforum.org] On Behalf Of Allegro
> Sent: Monday, January 03, 2005 9:14 PM
> To: geometry-puzzles@mathforum.org
> Subject: Re: Generalization of the Pyphagorean Theorem?
>
> On 3 Jan 05 07:45:01 -0500 (EST), Zak Seidov wrote:
> > "Generalization" of the Pythagorean Theorem?
> >
> >For a right triangle ABC, with right angle C,
> >the area of regular triangle ABC'
> >built on AB (hypotenuse)
> >equals to sum of area of regular triangle BCA'
> >built on leg BC and
> >area of regular triangle ACB'
> >built on leg AB:
> >
> >S_ABC'=S_BCA'+S_ACB'.
> >
> >
> > *A
> > ^ |\+ ++
> > ^ | \ ++
> > ^ | \ + * C'
> > B'* | \ +
> > ^ | \ +
> > ^ | \ +
> > ^ |______\+
> > C B
> > - -
> > - -
> > * A'
> >
> >see:
> ><a
>
href="http://www.geocities.com/zseidov/GenPhytTheo.html"><a href="http://www.geoc">http://www.geoc</a>
it
> ies.com/zseidov/GenPhytTheo.html</a>
> >
> >
> >"Regular triangles" may be replaced by any
> >"regular polygones"
> >or even by any similar figures,
> >simplest of the last being
> >isosceles triangles with bases as AB, BC, and AC,
> >and equal angles at vertices A', B' and C'.
Happy new year.
The generalization you mention is not
something new, and rather well known.
It first appearence is in ancient Greek
mathematics, in the 3rd century B.C. Indeed
you can find it in Book V of Euclid's Elements.
There Euclid generilizes the standard Pythagorean
theorem he proved in Book I.
When he comes to Book V (attributed to Eudoxos) he
has the so called "theory of proportion". The basis
of his generalization is the (standard by now)
theorem that similar figures are as the squares
of their corresponding sides. For circles the
relevant theorem is in his Book XII.
All the best
Michael Lambrou
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