The following proof of the Pythagorean Theorem was discovered in by President J.A Garfield in 1876.
The Pythagorean Theorem states that for any right triangle of sides a, b and c, where c is the hypotenuse, c2 = a2 + b2
The formula for the area of a trapezoid is: half the sum of the bases times the height.
Therefore, the area of the trapezoid in the diagram above can be calculated as:
(a + b)/2 · (a + b)
However, the area can also be calculated as the sum of the three triangles:
ab/2 + ab/2 + c · c/2
(a + b)/2 · (a + b) = ab/2 + ab/2 + c · c/2
which simplifies to:
(a + b)2 = 2ab + c2
a2 + b2 + 2ab - 2ab = c2
c2 = a2 + b2