# 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

 $\displaystyle\frac{(a+b)^{2}}{2}$ $\displaystyle=2\frac{ab}{2}+\frac{c^{2}}{2}$ $\displaystyle\frac{a^{2}+2ab+b^{2}}{2}$ $\displaystyle=ab+\frac{c^{2}}{2}$ $\displaystyle a^{2}+2ab+b^{2}$ $\displaystyle=2ab+c^{2}$ $\displaystyle a^{2}+b^{2}$ $\displaystyle=c^{2}$
Title Garfield’s proof of Pythagorean theorem GarfieldsProofOfPythagoreanTheorem 2013-03-22 17:09:33 2013-03-22 17:09:33 rm50 (10146) rm50 (10146) 10 rm50 (10146) Proof msc 51-00