Garfield's Proof of the Pythagorean Theorem

measure of angle ABC

0.5

step

1

2

3

4

5

This Demonstration shows President James Garfield's elegant proof of the Pythagorean theorem.

The area of the trapezoid

CADE

(step 5) can be calculated in two ways: as a trapezoid with base

a+b

and heights

a

and

b

, or as the sum of the areas of three triangles:

(a+b)(a+b)/2=

2

c

/2+2ab/2

.

Multiply through by 2 and expand the left-hand side to get

2

a

+2ab+

2

b

=

2

c

+2ab

,

which simplifies to

2

a

+

2

b

=

2

c

.

Details

James Abram Garfield (1831–1881), the twentieth president of the United States, published this proof in the New England Journal of Education.

References

[1] S. Klebe, "Garfield, the Pythagorean Theorem, and the Fight for Universal Education," Executive Intelligence Review, 22(9), 1995 pp. 50–51. www.larouchepub.com/eiw/public/1995/eirv22n09-19950224/eirv22n09-19950224_050-garfield_the_pythagorean_theorem.pdf.

Permanent Citation

Alvaro José Ibarra Rivas

"Garfield's Proof of the Pythagorean Theorem"
http://demonstrations.wolfram.com/GarfieldsProofOfThePythagoreanTheorem/
Wolfram Demonstrations Project
Published: March 20, 2013