Euclid,

ElementsI 38©

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

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Prop. 38: Triangles that are on equal bases and are in the same parallels are equal to one another.

(diagram 1) Let there be triangles, ABG, DEZ on equal bases BG, EZ, and in the same parallels, BZ, AD. I say that triangle ABG is equal to triangle DEZ. (diagram 2) For let AD be extended on both sides to H, Q, (diagram 3) and let through B a parallel to GA be drawn, GH, and let through Z a parallet to DE be drawn, ZQ. (diagram 4) Therefore, each of HBGA, DEZQ is a parallelogram. And HBGA is equal to DEZQ. For they are on equal bases, BG, EZ, and are in the same parallels, BZ, HQ. And triangle ABG is half of parallelogram HBGA (for diameter AB bisects it), while triangle ZED is half of parallelogram DEZQ (for diameter DZ bisects it). But halves of equals are equal to one another. Therefore, triangle ABG is equal to triangle DEZ. Therefore, triangles that are on equal bases and are in the same parallels are equal to one another, just what it was required to show.

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