Triangles which are on equal bases and in the same parallels are equal to one another.
在相等底边、同两条平行线之间的三角形彼此相等。
三角形 ABC 与 DEF 在相等底 BC、EF 上，夹在两条相同的平行线 GH 与底之间。BG、FH 辅助平行段，使两三角形面积相等。
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Let ABC, DEF be triangles on equal bases BC, EF and in the same parallels BF, AD; I say that the triangle ABC is equal to the triangle DEF. For let AD be produced in both directions to G, H; through B let BG be drawn parallel to CA, [I. 31] and through F let FH be drawn parallel to DE.
等底同平行线间的两个三角形，各自补成对应平行四边形。
Then each of the figures GBCA, DEFH is a parallelogram; and GBCA is equal to DEFH; for they are on equal bases BC, EF and in the same parallels BF, GH. [I. 36] Moreover the triangle ABC is half of the parallelogram GBCA; for the diameter AB bisects it.
由 euclid-elements/book1-prop-036，这些平行四边形相等。
[I. 34] And the triangle FED is half of the parallelogram DEFH; for the diameter DF bisects it. [I. 34] [But the halves of equal things are equal to one another.] Therefore the triangle ABC is equal to the triangle DEF.
再用 euclid-elements/book1-prop-034，每个三角形是对应平行四边形的一半。
Therefore etc.
相等整体的一半相等，所以两个三角形相等。