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Garfield's proof of the Pythagorean theorem is an original proof of the
Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
discovered by
James A. Garfield James Abram Garfield (November 19, 1831 – September 19, 1881) was the 20th president of the United States, serving from March 1881 until his death in September that year after being shot two months earlier. A preacher, lawyer, and Civi ...
(November 19, 1831 – September 19, 1881), the 20th
president of the United States The president of the United States (POTUS) is the head of state and head of government of the United States. The president directs the Federal government of the United States#Executive branch, executive branch of the Federal government of t ...
. 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 a 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 when he died following his assassination in July. 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. (A collection of 370 different proofs of the Pythagorean theorem.)


The proof

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\text ACED & = \text\times \text\\ & = CE\times\tfrac(AC+DE)=(a+b)\times \tfrac(a+b)\end. :\begin \text ACED & = \text\Delta ACB + \text \Delta ABD + \text \Delta BDE \\& = \tfrac(a\times b) + \tfrac(c\times c) + \tfrac(a\times b)\end From these one gets :(a+b)\times \tfrac(a+b) = \tfrac(a\times b) + \tfrac(c\times c) + \tfrac(a\times b) which on simplification yields :a^2+b^2=c^2


References

{{reflist Pythagorean theorem Articles containing proofs
Theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...
Euclidean plane geometry History of geometry Theorems in plane geometry James A. Garfield