Euclid, Elements I 37©
translated by Henry Mendell (Cal. State U., L.A.)

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Prop. 37: Triangles that are on the same base and are in the same parallels are equal to one another.

(general diagram)

Diagrams for book 1 proposition 37, following the display, construction, and demonstration(diagram 1) Let there be triangles, ABG, DBG on the same base, BG, and in the same parallels, AD, BG. I say that triangle ABG is equal to triangle DBG. (diagram 2) Let AD be extended on both sides to E, Z, (diagram 3) and through B, let a parallel to GA be drawn, BE, (diagram 4) and through G let a parallel to BD be drawn, GZ. (diagram 5) Therefore, each of EBGA, DBGZ is a parallelogram. And they are equal. For they are both on the same base, BG and are in the same parallels, BG, EZ. (diagram 6) And triangle ABG is half of parallelogram EBGA (for AB bisects it), (diagram 7) while triangle DBG is half of parallelogram DBGZ. (for diameter DG bisects it). But halves of equals are equal to one another. (diagram 8) Therefore, triangle ABG is equal to triangle DBG. Therefore, triangles that are on the same base and are in the same parallels are equal to one another, just what it was required to show.

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