# A US President’s Proof of the Pythagorean Theorem

James Garfield, the 20th president of the United States, came up with an original proof of the Pythagorean Theorem in 1876 when he was still a Congressman. His proof was published in *New England Journal of Education*.

Recall that the Pythagorean Theorem states that given a right triangle with sides , , and hypotenuse , the following equation is always satisified:

.

President Garfield’s proof is quite simple. We can do this in three steps:

- Find the area of figure above using the trapezoid
- Find the area of the same figure using the three triangles
- Equate the results in 1 and 2.

Proof:

(1) Finding the area of the figure using the trapezoid

:

:

:

.

(2) Finding the area of the figure using the triangles

area of the red triangle =

area of the green triangle =

area of the blue triangle =

total area

(3) The areas calculated in (1) and (2) are equal, therefore we can equate (1) and (2).

Multiplying both sides by , we have .

Subtracting from both sides, we have . That completes the proof.