Garfield's proof of the Pythagorean theorem

Garfield in 1881

Garfield's proof of the Pythagorean theorem is an original proof the Pythagorean theorem invented by James A. Garfield (November 19, 1831 – September 19, 1881), the 20th president of the United States. The proof appeared in print in the New-England Journal of Education (Vol. 3, No.14, April 1, 1876). At the time of the publication of the proof Garfield was not the President, he was only the Congressman from Ohio. He assumed the office of President on March 4, 1881, and served in that position only for a brief period up to September 19, 1881. Garfield was the only President of the United States to have contributed anything original to mathematics. The proof is nontrivial and, according to the historian of mathematics, William Dunham, "Garfield's is really a very clever proof." The proof appears as the 231st proof in The Pythagorean Proposition, a compendium of 370 different proofs of the Pythagorean theorem.

The proof

Diagram to explain Garfield's proof of the Pythagorean theorem

In the figure, $ABC$ is a right-angled triangle with right angle at $C$. The side-lengths of the triangle are $a,b,c$. Pythagorean theorem asserts that $c^{2}=a^{2}+b^{2}$.

To prove the theorem, Garfield drew a line through $B$ perpendicular to $AB$ and on this line chose a point $D$ such that $BD=BA$. Then, from $D$ he dropped a perpendicular $DE$ upon the extended line $CB$. From the figure, one can easily see that the triangles $ABC$ and $BDE$ are congruent. Since $AC$ and $DE$ are both perpendicular to $CE$, they are parallel and so the quadrilateral $ACED$ is a trapezoid. The theorem is proved by computing the area of this trapezoid in two different ways.

${\begin{aligned}{\text{area of trapezoid }}ACED&={\text{height}}\times {\text{average of parallel sides}}\\&=CE\times {\tfrac {1}{2}}(AC+DE)=(a+b)\times {\tfrac {1}{2}}(a+b)\end{aligned}}$.

${\begin{aligned}{\text{area of trapezoid }}ACED&={\text{area of }}\Delta ACB+{\text{area of }}\Delta ABD+{\text{area of }}\Delta BDE\\&={\tfrac {1}{2}}(a\times b)+{\tfrac {1}{2}}(c\times c)+{\tfrac {1}{2}}(a\times b)\end{aligned}}$

From these one gets

$(a+b)\times {\tfrac {1}{2}}(a+b)={\tfrac {1}{2}}(a\times b)+{\tfrac {1}{2}}(c\times c)+{\tfrac {1}{2}}(a\times b)$

which on simplification yields

$a^{2}+b^{2}=c^{2}$