President Garfield's Trapezoid Proof

Garfield's proof builds a trapezoid from two copies of a right triangle, with legs a and b and hypotenuse c, plus a right isosceles triangle with legs c. The trapezoid's area is 1 over 2(a + b)(a + b); the three triangles' areas add to ab + 1 over 2c². Setting them equal gives a² + b² = c². James A. Garfield published it in 1876.

Building the trapezoid

The whole proof lives in one figure, so it pays to build it carefully. Start with a right triangle ACB with the right angle at C, leg CA = a standing up, leg CB = b lying flat, and hypotenuse AB = c. (For the names of the parts, see right triangles.)

1. Extend the base. Continue the line CB past B to a point E with BE = a. Now CE = a + b.
2. Raise the second triangle. At E, draw ED perpendicular to CE, on the same side as A, with length ED = b. Triangle BED has legs a and b with a right angle between them, so by SAS it is congruent to triangle ACB, and its hypotenuse is BD = c. In the animation this copy is made by turning the first triangle a quarter turn.
3. Close the shape. Join A to D. The sides CA and ED are both perpendicular to CE, so they are parallel, and ACED is a trapezoid with parallel sides a and b and height a + b.

Some books describe the construction in a different order, starting with a line through B perpendicular to AB. Every version produces this same figure.

Why angle ABD is a right angle

The trapezoid is cut into three triangles: ACB, BED and the middle one, ABD. To find the middle triangle's area, we need to know its angle at B.

Call the acute angles of the original triangle α (opposite a) and β (opposite b). In triangle ACB, the angle at B is opposite the side CA = a, so ∠ABC = α. In triangle BED, the angle at B is opposite the side ED = b, so ∠DBE = β. The points C, B and E lie on one line, so the three angles at B add to 180°:

In a right triangle the two acute angles add to 90°, so α + β = 90° and ∠ABD = 90°. Triangle ABD is therefore a right triangle with both legs equal to c, and its area is 1 over 2c · c = 1 over 2c².

The area, two ways

As a trapezoid. The area of a trapezoid is half the sum of the parallel sides times the distance between them. Here the parallel sides are a and b, and the distance between them is CE = a + b:

As three triangles. The two congruent triangles have area 1 over 2ab each, and the middle triangle has area 1 over 2c²:

Set them equal. Multiply both sides by 2, expand (a + b)², and subtract 2ab:

The complete proof

Half of a famous picture

Garfield's trapezoid is exactly half of the square used in the algebraic proof. Take a square of side a + b with four triangles around a tilted square of side c, and cut it along a diagonal of the tilted square.

The cut passes through two opposite corners of the tilted square, so it halves the tilted square into two of Garfield's middle triangles, and each half of the big square contains two of the four corner triangles. Doubling Garfield's equation, 1 over 2(a + b)² = ab + 1 over 2c², gives (a + b)² = 4 · 1 over 2ab + c², which is the algebraic proof. So Garfield's proof is the same argument done with half of the figure. Whether Garfield knew the full-square version isn't recorded. The full square, in turn, is the first arrangement of the rearrangement proof.

The history: a proof from Congress

James Abram Garfield (1831–1881) is the only U.S. president known to have published an original proof in mathematics. Before politics he studied at Williams College, graduating in 1856, and taught at the Western Reserve Eclectic Institute in Hiram, Ohio, now Hiram College, where he also served as principal. He was a Union general in the Civil War and then represented Ohio in the U.S. House of Representatives from 1863 to 1880.

The proof appeared in the New-England Journal of Education on April 1, 1876 (volume 3, number 14, page 161), under the heading "Pons Asinorum". The editors explained that it had been "hit upon in some mathematical amusements and discussions with other M. C.'s", meaning members of Congress, and added: "we think it something on which the members of both houses can unite without distinction of party." Garfield's diary from March 1876 records him showing the proof to a Dartmouth mathematics professor.

The heading is curious: "pons asinorum", the "bridge of asses", is usually the nickname of a different theorem, Euclid's Proposition I.5, which says that the base angles of an isosceles triangle are equal. Garfield was elected president in 1880 and took office in March 1881. He was shot on July 2, 1881, and died of his wounds on September 19, 1881. Elisha Loomis's collection of proofs, The Pythagorean Proposition, lists his proof as number 231. For the wider story, see the history of the theorem or the list of all the proofs on this site.

Using it in class

Garfield's proof is a favorite with teachers because it uses only the trapezoid area formula and one algebraic expansion, and the story gives students a reason to care. A good sequence is to have students draw the figure themselves from the three construction steps, find ∠ABD on their own, and only then compute the area two ways. It pairs naturally with the algebraic proof: ask students to explain why one is "half" of the other. A complete lesson that uses it is on the lesson plan page, more hands-on ideas are among the classroom activities, and the teaching guide shows where proofs fit in a unit.