Euclid,

ElementsI 46©

translated by Henry Mendell (Cal. State U., L.A.)

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Prop. 46: From the given straight-line, to describe up a square.

(diagram 1) Let the given straight line be AB. It is required, in fact, to describe up a square from straight-line AB. (diagram 2) Let AG be drawn from straight-line AB from the point on it, A, (diagram 3) and let AD be positioned equal to AB. (diagram 4) And through point D let a parallel, DE, to AB be drawn, (diagram 5) and through point B let a parallel, BE, to AD be drawn. (diagram 6) Therefore, ADEB is a parallelogram. Therefore, AB is equal to DE, and AD to BE. (diagram 7) But AB is equal to AD. Therefore, four, BA, AD, DE, EB, are equal to one another. Therefore, parallelogram ADEB is equilateral. (diagram 8) I say, in fact, that it is also right-angled. For since AD falls into parallels, AB, DE, therefore, the angles by BAD, ADE are equal to two right-angles. (diagram 9) But the angle by BAD is right. Therefore, the angle by ADE is also right. (diagram 10) But the opposite sides and angles of parallelogram regions are equal to one another. (diagram 11) Therefore, each of the opposite angles by ABE, BED is also right. Therefore ADEB is a rectangle. But it was also shown equilateral. Therefore, it is a square. And it is described up from straight-line AB, just what it was required to make.

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