Triangles which are on the same base and in the same parallels are equal to one another.
在同一底边、同两条平行线之间的三角形彼此相等。
三角形 ABC、DBC 共底 BC；A、D 在过 BC 平行线 EF 上。BE、CF 为辅助平行段，证明两三角形面积相等。
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Let ABC, DBC be triangles on the same base BC and in the same parallels AD, BC; I say that the triangle ABC is equal to the triangle DBC. Let AD be produced in both directions to E, F; through B let BE be drawn parallel to CA, [I. 31] and through C let CF be drawn parallel to BD.
同底同平行线间的两个三角形，各自补成同底同平行线间的平行四边形。
[I. 31] Then each of the figures EBCA, DBCF is a parallelogram; and they are equal, for they are on the same base BC and in the same parallels BC, EF. [I. 35] Moreover the triangle ABC is half of the parallelogram EBCA; for the diameter AB bisects it.
由 euclid-elements/book1-prop-035，这两个平行四边形相等。
[I. 34] And the triangle DBC is half of the parallelogram DBCF; for the diameter DC bisects it. [I. 34] [But the halves of equal things are equal to one another.] Therefore the triangle ABC is equal to the triangle DBC.
每个平行四边形都由对角线分成两个相等三角形（euclid-elements/book1-prop-034）。
Therefore etc.
所以原来两个三角形相等。