Euclid,

ElementsI 19©

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

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Prop. 19: Subtending the larger angle of any triangle is the larger side.

(diagram 1) Let there be a triangle, ABG, having the angle by ABG larger than that by BGA. (diagram 2) I say that a side, AG, is also larger than a side, AB. For if not, either AG is equal to AB or is smaller. (diagram 3) And so AG is not equal to AB. For an angle, that by ABG, would also be equal to that by AGB. But it is not. Therefore, AG is not equal to AB. (diagram 4) Truly, AG is not smaller than AB either. For an angle, that by ABG, would also be smaller than that by AGB. But it is not. (diagram 2)Therefore, AG is not smaller than AB. But it was shown that it is not equal either. Therefore, AG is larger than AB. Therefore, subtending the larger angle of any triangle is the larger side, just what it was required to show.

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