Euclid, Elements I 10©
translated by Henry Mendell (Cal. State U., L.A.)
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Prop. 10: To bisect the given finite straight-line.
(diagram 1) Let the given finite straight-line be AB. It is, in fact, required to bisect finite straight-line AB. (diagram 2) Let an equilateral triangle, ABG, be constructed on it, (diagram 3) and let angle AGB be bisected by straight-line GD. I say that AB has been bisected at point D. (diagram 4) For since AG is equal to GB, but GD is common, in fact two, AG, GD, are respectively equal to two, BG, GD. And an angle, that by AGD is equal to an angle, that by BGD. (diagram 5) Therefore, a base, AD, is equal to a base, BD. Therefore, the given finite straight-line has been bisected at D, just what it was required to make.