Euclid, Elements I 12©
translated by Henry Mendell (Cal. State U., L.A.)

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Prop. 12: To draw a perpendicular straight line to the given infinite straight-line from the given point, which is not one it.

(general diagram)

Diagrams for book 1 proposition 12, following the display, construction, and demonstration(diagram 1) Let the given infinite straight-line be AB, the given point, which is not on it, G. It is, in fact, required to draw a perpendicular straight-line to the given infinite straight-line, AB, from the given point, G, which is not on it. (diagram 2) For let there be taken on the other sides of straight-line AB a point as happens, D, (diagram 3) and with a center, G, and distance, GD, let a circle be described, EZH, (diagram 4) and let straight-line EH be bisected at Q. (diagram 5) And let straight-lines GH, GQ, GE be joined. I say that a perpendicular, GQ, has been drawn to the given infinite straight-line, AB, from the given point, G, which is not on it. (diagram 6) For since HQ is equal to QE and QG is common, in fact two, HQ, QG, are respectively equal to two, EQ, QG. And a base, GH, is equal to a base, GE. (diagram 7) Therefore, an angle, that by GQH, is equal to an angle, that by EQG. And they are in succession. But whenever a straight-line stood on a straight-line makes successive angles equal to one another, each of the equal angles is right, and the straight-line stood-on it is called perpendicular to what it stands on. Therefore, a perpendicular, GQ, has been drawn to the given infinite straight-line, AB, from the given point, G, which is not on it, just what it was required to make.

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