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Below is a collection of 118 approaches to proving the theorem.Many of the proofs are accompanied by interactive Java illustrations. C.) or someone else from his School was the first to discover its proof can't be claimed with any degree of credibility. C.) by an early 20th century professor Elisha Scott Loomis.Then he asked, "Suppose these three squares were made of beaten gold, and you were offered either the one large square or the two small squares. " Interestingly enough, about half the class opted for the one large square and half for the two small squares.Both groups were equally amazed when told that it would make no difference.Which says that the area AC² of the square on side AC equals the area of the rectangle AELM. Calderhead ( We start with two squares with sides a and b, respectively, placed side by side. The construction did not start with a triangle but now we draw two of them, both with sides a and b and hypotenuse c.Similarly, the area BC² of the square on side BC equals that of rectangle BMLD. Note that the segment common to the two squares has been removed.

The proof below is a somewhat shortened version of the original Euclidean proof as it appears in Sir Thomas Heath's translation. This is because, and ∠BAF = ∠BAC ∠CAF = ∠CAB ∠BAE = ∠CAE.We can compute the area of the big square in two ways.Thus (a b)² = 4·ab/2 c² simplifying which we get the needed identity.Three of these have been rotated 90°, 180°, and 270°, respectively. Let's put them together without additional rotations so that they form a square with side c.The square has a square hole with the side and a hole with the side c.

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ΔABF has base AF and the altitude from B equal to AC.