Euclid,

ElementsI 20©

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

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Prop. 20: The two sides of any triangle, taken in any way, are larger than the remaining.

(diagram 1) For let there be a triangle, ABG. I say that the two sides of triangle ABG, taken in any way, are larger than the remaining side, BA, AG than BG and AB, BG than AG and BG, GA than AB. (diagram 2) For let BA be drawn through to point D, and let an equal, AD, to GA be positioned, (diagram 3) and let DG be joined. (diagram 4) And so, since DA is equal to AG, an angle, that by ADG, is also equal to that by AGD. (diagram 5) Therefore, the angle by BGD is larger than that by ADG. (diagram 6) And since DGB is a triangle having the angle by BGD larger than that by BDG, and the larger side subtends the larger angle, therefore, DB is larger than BG. But DA is equal to AG. Therefore, BA, AG are larger than BG. Similarly, in fact, we will show that AB, BG are also larger than GA and BG, GA than AB. Therefore, the two sides of any triangle, taken in any way, are larger than the remaining, just what it was required to show.

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