Garfield's proof of the Pythagorean theorem is an original proof of thePythagorean theorem discovered byJames A. Garfield (November 19, 1831 – September 19, 1881), the 20thpresident of the United States. The proof appeared in print in theNew-England Journal of Education (Vol. 3, No.14, April 1, 1876).^[1][2] At the time of the publication of the proof Garfield was acongressman fromOhio. He assumed the office of President on March 4, 1881, and served in that position until his death on September 19, 1881, having succumbed to injuries sustained when he wasshot in an assassination in July.^[3] Garfield is thus far the only President of the United States to have contributed anything original to mathematics. The proof is nontrivial and, according to the historian of mathematicsWilliam Dunham, "Garfield's is really a very clever proof."^[4] The proof appears as the 231st proof inThe Pythagorean Proposition, a compendium of 370 different proofs of the Pythagorean theorem.^[5]

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

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

.

From these one gets

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

which on simplification yields

${\displaystyle a^2+b^2=c^2.}$

Garfield's proof is a variant of one of thealgebraic proofs (pictured at right), but using only half of the diagram. The pictured version observes that the area of the outer square equals the area of the inner square plus four congruent triangles, which is to say

${\displaystyle (a+b)\times (a+b)=(c\times c)+4\times \frac{1}{2}(a\times b)}$

and simplifies the same way.

• Pythagorean theorem
• Pythagoras
• Euclidean plane geometry
• History of geometry
• Theorems in plane geometry
• James A. Garfield
• 19th century in mathematics
• 1876 in science

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