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 $a$, $b$, and hypotenuse $c$, the following equation is always satisified:

$a^2 + b^2 = c^2$.

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

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

Proof:

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

$b_1$: $a$

$b_2$: $b$

$h$: $a + b$

$A = \displaystyle\frac{1}{2}h(b_1 + b_2)$

$A = \displaystyle \frac{1}{2}(a+b)(a+b) = \frac{a^2}{2}+ab+\frac{b^2}{2}$.

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

area of the red triangle = $\displaystyle\frac{ab}{2}$

area of the green triangle =  $\displaystyle\frac{c^2}{2}$

area of the blue triangle = $\displaystyle\frac{ab}{2}$

total area $A = \displaystyle\frac{ab}{2} + \frac{c^2}{2} + \frac{ab}{2} = ab + \frac{c^2}{2}$

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

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

Multiplying both sides by $2$, we have $a^2 + 2ab + b^2 = 2ab + c^2$.

Subtracting $2ab$ from both sides, we have $a^2 + b^2 = c^2$. That completes the proof.

5 thoughts on “A US President’s Proof of the Pythagorean Theorem”

1. I find this proof interesting. It is obviously the standard ‘square proof’ – in which the green triangle above is half of what is usually a square. Even though it is of course different to the proof I’m referring to, it is ultimately based on the same idea. Still, great to think that a president was attempting his own proofs!

• You’re right. Doubling the figure and putting them together is the ‘square proof.’ However, what is interesting is that President Garfield did not take up a math-related course.