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.
(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 =
(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.