a2 + b2 = c2
Of course it has a direct geometric formulation.
Click on & move the node to change the shape of the triangle.
For many of us, this is the first result in geometry that does not seem to be self-evident. This has apparently been a common experience throughout history, and proofs of this result, of varying rigour, have appeared early in several civilizations. We present a selection of proofs, dividing roughly into three types, depending on what geometrical transformations are involved.
The oldest known proof
Proofs that use shears (including Euclid's). These work because shears of a figure preserve its area. Some of these proofs use rotations, which are also area-preserving.
Proofs that use translations. These dissect the large square into pieces which one can rearrange to form the smaller squares. Some of these are among the oldest proofs known.
Proofs that use similarity. These are in some ways the simplest. They rely on the concept of ratio, which although intuitively clear, in a rigourous form has to deal with the problem of incommensurable quantities (like the sides and the diagonal of a square). For this reason they are not as elementary as the others.
Thomas L. Heath, The thirteen books of Euclid's Elements in three volumes, Dover, 1956. The Elements of Euclid are available on the Internet as are all of Heath's comments.
Elisha Loomis, The Pythagorean Proposition, National Council of Teachers of Mathematics, 1968. This eccentric book was first compiled in 1907, first published in 1928 (at a price of $2.00!), and reissued in this edition. It contains 365 more or less distinct proofs of Pythagoras' Theorem. The total effect is perhaps a bit overwhelming, and the quality of the figures is very poor, but nonetheless there are a few gems distributed throughout.
From the life of Thomas Hobbes in John Aubrey's Brief lives, about 1694.
From pp. 9-11 in the opening autobiographical sketch of Albert Einstein: Philosopher-Scientist, edited by Paul Arthur Schilpp, published in 1951.
"You mentioned isosceles triangles. Will it do if I prove Pythagoras for you?"
"Jesus," he said. "The square on the hypotenuse. I'll bet you can't."
I did it with a bayonet, on the earth beside my pit - which may have been how Pythagoras himself did it originally, for all I know. I went wrong once, having forgotten where to drop the perpendicular, but in the end there it was ... He folowed it so intently that I felt slightly worried; after all, it's hardly normal to be utterly absorbed in triangles and circles when the surrounding night may be stiff with Japanese.
From p. 150 of Quartered safe out here by George MacDonald Fraser, an account of his experiences in the Burmese fighting in 1944-45 against the Japanese. Published by Harper-Collins in 1992.