Euclid, Elements I 36©
translated by Henry Mendell (Cal. State U., L.A.)
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Prop. 36: Parallelograms that are on equal bases and are in the same parallels are equal to one another.
(diagram 1) Let there be parallelograms, ABGD, EZHQ that are on equal bases, BG, ZH, and are in the same parallels, AQ, BH. I say that parallelogram ABGD is equal to EZHQ. (diagram 2) For let BE, GQ be joined. (diagram 3) And since BG is equal to ZH, but ZH is equal to EQ, therefore, BG is also equal to EQ. But they are also parallel. And EB, QG join them. (diagram 4) But straight-lines joining equals and parallels on the same sides are equal and parallel. Therefore, EB QG are equal and parallel. (diagram 5) Therefore, EBGQ is a parallelogram. (diagram 6) It is also equal to ABGD. For they have a base that’s the same, BG, and are in the same parallels, BG, AQ. (diagram 7) For the same reasons, in fact, EZHQ is also equal to the same, EBGQ. (diagram 8) Thus, parallelogram ABGD is also equal to EZHQ. Therefore, parallelograms that are on equal bases and are in the same parallels are equal to one another, just what it was required to show.