
Re: [HM] "President Garfield's Proof"
Posted:
Jan 15, 2000 10:03 AM


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 fixedpoint 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.
Clark Kimberling

