A US president's proof

2026-05-11

Take two copies of any right triangle. Lay one flat. Stand the other up next to it, hypotenuses meeting at a right angle. Connect the tops. You have a trapezoid - and the proof has already happened.

The two coloured triangles are identical, just rotated. The yellow triangle in the middle is what's left over - and it has to be a right triangle too, because the two hypotenuses meet at 90°. Its legs are both c, the original hypotenuse. So its area is ½c².

Now look at the outline. The whole shape is a trapezoid with two parallel vertical sides - one of length a, one of length b - and a base of length a + b. The area of any trapezoid is the average of its parallel sides times the distance between them: ½(a + b) × (a + b), which is ½(a + b)².

So the same trapezoid has two areas:

½(a + b)²  =  ½ab + ½ab + ½c²

Multiply both sides by 2 and expand the left:

a² + 2ab + b²  =  2ab + c²

The 2ab on each side cancels, and what's left is a² + b² = c². That is the theorem.

Why the middle triangle has to be right-angled

Here is the only place the proof asks you to pause. The two coloured triangles share a vertex at the bottom of the figure - the hinge. The hypotenuse of the first leaves that hinge going up and to the left. The hypotenuse of the second leaves it going up and to the right. Are those two hypotenuses perpendicular?

The angles at the hinge are: the right angle of one triangle, the acute angle from the first triangle, and the acute angle from the second triangle. Those three together fill the straight line of the base, so they add to 180°. The right angle takes 90°. The two acute angles of any right triangle add to 90°. So the middle angle - the one between the two hypotenuses - is exactly 90°. The middle triangle is right-angled, with both legs equal to c.

The president

James A. Garfield, US Representative from Ohio, published this proof in the New England Journal of Education in 1876, five years before he became the 20th president of the United States. He is apparently the only US president to have contributed to mathematics. The proof had been discovered earlier in other forms, but this packaging - the trapezoid that quietly does all the work - is his.

The same shape, the same area, written two ways. The cross-terms cancel, and what's left is the theorem.

Other proofs of a² + b² = c²: Bhāskara's rearrangement.