A US President’s Proof of the Pythagorean Theorem

High School Mathematics

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.

Pythagorean Theorem proof by President Garfield

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.

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