By Carlos J. Moreno
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Additional resources for Advanced Analytic Number Theory, Part I: Ramification Theoretic Methods (Contemporary Mathematics,Volume 15)
Straight-lines) DE, EF , respectively. And the angles they encompass (are also equal). Thus, the base AH is equal to the base DF , and the triangle ABH is equal to the triangle DEF , and the remaining angles subtended by the equal sides will be equal to the (corresponding) remaining angles [Prop.
34]. And triangle F ED (is) half of parallelogram DEF H. For the diagonal DF cuts the latter in half. ] Thus, triangle ABC is equal to triangle DEF . Thus, triangles which are on equal bases and between the same parallels are equal to one another. (Which is) the very thing it was required to show. Ð . Proposition 39 Τὰ ἴσα τρίγωνα τὰ ἐπὶ τῆς αὐτῆς βάσεως ὄντα καὶ ἐπὶ Equal triangles which are on the same base, and on τὰ αὐτὰ μέρη καὶ ἐν ταῖς αὐταῖς παραλλήλοις ἐστίν. the same side, are also between the same parallels.