Pythagorean Theorem: Euclid's proof

The Pythagorean Theorem (and its inverse) completes book one of Euclid’s “Ele­ments”. Propo­si­tion XLVII (47) states: In right trian­gles the square on the side enclosing the right angle is equal to <taken together> the squares on the sides enclosing the right angle.

The familiar term “hypotenuse” comes (via Latin) from the Greek: “sub­tending the right angle” is a literal trans­la­tion of the text of “Ele­ments” — ἡ τὴν ὀρθὴν γωνίαν ὑποτείνουσα.

An elegant and elemen­tary proof of the Pythagorean theorem of the type “Look!”, essen­tially anal­o­gous to that given in “Ele­ments”, may be presented in pictures. Let us divide the square built on the hypotenuse into two parts by the contin­u­a­tion of the alti­tude of the right triangle dropped from the vertex of the right angle. It turns out that the smallest of the resulting rectan­gles is equal in area to the square built on the smaller cathetus, and the larger is equal to the square built on the larger cathetus.

In the vari­a­tions of Euclid’s proof below, the areas of the figures do not change when they are skewed: their bases and heights always remain constant.



When paral­lel­o­grams are rotated with respect to the vertices of acute angles, their sides will lie on the height because their vertices will be at the vertex of the right angle. Indeed, the side of a small square is rotated 90 degrees and goes to the side of a triangle; and the long sides of paral­lel­o­grams are parallel to the sides of the square on the hypotenuse. The differ­ence from the proof given in “Ele­ments” and called “wind­mill proof” is that Euclid did not use anima­tion and consid­ered not the paral­lel­o­grams them­selves, but the trian­gles which are their halves.



Another similar proof is to extend the sides of the small squares until they inter­sect.



With such a proof, one has to think about and justify why the point of inter­sec­tion turns out to lie on the contin­u­a­tion of the alti­tude.