A US President’s Proof of the Pythagorean Theorem

garfield-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.

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4 Comments

  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!

    Reply
    • 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.

      Reply

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