Equal triangles which are on equal bases and on the same side are also in the same parallels.
等底同侧的相等三角形，其顶点在同一条与底边平行的直线上。
等积三角形 ABC、DEF 在相等底 BC、EF 上同侧；要证 A、D 两顶点在同一条与底平行的直线上。
正文图形由校订坐标生成；点、线、角、圆可与证明和问答联动。
Let ABC, CDE be equal triangles on equal bases BC, CE and on the same side. I say that they are also in the same parallels.
把等底相等三角形放在同侧。
For let AD be joined; I say that AD is parallel to BE. For, if not, let AF be drawn through A parallel to BE [I. 31], and let FE be joined.
若顶点不在同一条与底平行的直线上，则用等底构造相同平行线间的比较三角形。
Therefore the triangle ABC is equal to the triangle FCE; for they are on equal bases BC, CE and in the same parallels BE, AF. [I. 38] But the triangle ABC is equal to the triangle DCE; therefore the triangle DCE is also equal to the triangle FCE, [C.N. 1] the greater to the less: which is impossible.
由 euclid-elements/book1-prop-038 会推出面积不等。
Therefore AF is not parallel to BE. Similarly we can prove that neither is any other straight line except AD; therefore AD is parallel to BE.
矛盾说明两个顶点必须在同一条平行线上。