# Pythagoras…and the only US President to prove a theorem

The Pythagorean Theorem has been proved many times, but only once by a United States President. Five years before James A Garfield was elected president, he came up with a proof that involves a simple sheet of paper and some scissors.

Grab a piece of paper and cut two identical right angle triangles, trying to include one long side and one short side. Call the short side a, the long side b, and the hypotenuse c. Place them, opposing points together, on a piece of paper so that they look like this:

Draw the third line in to make three triangles. The total area of a triangle can be calculated as one half the base times the height. Each of the original triangles has an area of ½ ab, and the third has an area of ½c^2.

A trapezoid has an area calculated by its height times one half the sum of its uneven sides. So the trapezoid has an area of ½ (a+b)(a +b).

This simplifies to:
½c^2 = ½ a^2 + ½ b^2

Multiply it all by 2, and you get:
c^2 = a^2 + b^2

And that’s how to solve a maths problem, presidential-style.

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