James Garfield's Proof of the Pythagorean Theorem

James Garfield, the 20th President of the United States, gave the following simple proof of the Pythagorean theorem



It begins by constructing the trapezoid with parallel sides of height a and b and with base a + b. The area is given by the base times the average height or

A = (1/2)(a+b)².

On the other hand, we may compute the area of the three nonintersecting interior (and necessarily right) triangles

A = (1/2)(ab + ab + c²),

where c is the hypotenuse.
Setting up the equation for the two equal area calculations we have


(1/2)(a² + 2 a b + b²) = (1/2)(2 a b + c²),

or
a² + b² = c².