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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, c^{2} = a^{2} + b^{2}

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

Therefore:

(a + b)/2 · (a + b) = ab/2 + ab/2 + c · c/2

which simplifies to:

(a + b)^{2} = 2ab + c^{2}

Thus:

a^{2} + b^{2} + 2ab - 2ab = c^{2}

so:

c^{2} = a^{2} + b^{2}

Q.E.D.

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