Garfield’s proof of Pythagorean theorem

James Garfield, the ${20}^{\mathrm{th}}$ president of the United States, gave the following proof of the Pythagorean Theorem in 1876. Consider the following trapezoid (note that this picture is half of the diagram used in Bhaskara’s proof of the Pythagorean theorem (http://planetmath.org/ProofOfPythagoreasTheorem)).

Recall that the area of a trapezoid with two parallel sides (in this case, the left and right sides) $s_1$ and $s_2$ and height $h$ is

$$h\frac{s_1+s_2}{2}$$

So the area of the trapezoid above is

$$(a+b)\frac{a+b}{2}=\frac{{(a+b)}^2}{2}$$

The area of the yellow triangle (and that of the blue triangle) is

$$\frac{ab}{2}$$

while the area of the red triangle (also a right triangle) is

$$\frac{c^2}{2}$$

The two areas must be equal, so

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

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

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

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

Title	Garfield’s proof of Pythagorean theorem
Canonical name	GarfieldsProofOfPythagoreanTheorem
Date of creation	2013-03-22 17:09:33
Last modified on	2013-03-22 17:09:33
Owner	rm50 (10146)
Last modified by	rm50 (10146)
Numerical id	10
Author	rm50 (10146)
Entry type	Proof
Classification	msc 51-00