Subject: Re: [HM] "President Garfield's Proof"
From: Tom Apostol (apostol@caltech.edu)
Date: Sun Jan 16 2000 - 14:08:12 EST
At 10:03 AM 1/15/00 -0500, Clark Kimberling wrote:
>
> An HM reader has asked which proof of the Pythagorean theorem is the
> one attributed to Garfield. The answer may be of general interest.
>
> "Garfield's proof" goes with a figure consisting of three right
> triangles. Let ABC be the given one, with hypotenuse BA and shortest
> side BC. Draw right triangle ABD with length(BD)=length(BA) and D on
> the side of AB that doesn't contain C. Then draw right triangle BDE
> with length(DE)=length(BC) and E on the side of BD that doesn't
> contain A. Triangle BDE is congruent to triangle ABC. Let a,b,c
> denote the common sidelengths. To finish the proof, I'll quote
> Malcolm Graham's article, "President Garfield and the Pythagorean
> Theorem," in The Mathematics Teacher, Dec. 1976 (in a series for the
> American Bicentennial called "Events in the History of American
> Mathematics):
>
> "In figure 1, we see a trapezoid with bases a and b and height (a+b).
> The trapezoid is the union of three right triangles. Hence, the area
> of the trapezoid is equal to the sum of the areas of the three
> triangles.
>
> (a+b)(a+b)/2 = ab/2 + ab/2 + (c^2)/2
> (a+b)(a+b) = ab ab + c^2
> a^2 + 2ab + b^2 = 2ab + c^2
> a^2 + b^2 = c^2 ."
>
> In Elisha Scott Loomis's collection, The Pythagorean Proposition
> (National Council of Teachers of Mathematics, 1968) this proof is
> on page 231 and is also number 231 in the collection (a fixed-point
> theorem caught in action!)
>
> Eric Weisstein's remarkable CRC Concise Encyclopedia of Mathematics,
> 1999, on page 1465, includes "A novel proof ... discovered by James
> Garfield." I mention this to indicate that the "Garfield proof" is
> gaining in popularity. It has also been reproduced in other recent
> publications.
>
> Ah - even better: the same author's Mathworld ("The Web's Most
> Extensive Math Resource") offers the same coverage. Mathworld is
> at http://mathworld.wolfram.com . From there you can reach the
> "Garfield proof" by typing Garfield into the search box. Or, you can
> go directly to http://mathworld.wolfram.com/PythagoreanTheorem.html
> and scroll down aways.
>
Here's some additional information on the so-called Garfield proof.
On page 19 of the workbook/study guide for the video on The Theorem
of Pythagoras published by Project MATHEMATICS! in 1988 it is pointed
out that the diagram used in Garfield's proof is obtained by bisecting
a diagram used in a much older Chinese proof. In my view this proof
is not essentially different from the ancient Chinese proof, and it
has gained notoriety only because of Garfield's name.
Tom Apostol
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