Any prism which has a triangular base is divided into three pyramids equal to one another which have triangular bases.
任何以三角形为底的棱柱，可被分成三个彼此相等的三角底棱锥。
Let there be a prism in which the triangle ABC is the base and DEF its opposite; I say that the prism ABCDEF is divided into three pyramids equal to one another, which have triangular bases. For let BD, EC, CD be joined. Since ABED is a parallelogram, and BD is its diameter, therefore the triangle ABD is equal to the triangle EBD; [I. 34] therefore also the pyramid of which the triangle ABD is the base and the point C the vertex is equal to the pyramid of which the triangle DEB is the base and the point C the vertex.
连接BD、EC、CD。因ABED是平行四边形，BD为其对角线，故三角形ABD等于三角形EBD。
[XII. 5] But the pyramid of which the triangle DEB is the base and the point C the vertex is the same with the pyramid of which the triangle EBC is the base and the point D the vertex; for they are contained by the same planes. Therefore the pyramid of which the triangle ABD is the base and the point C the vertex is also equal to the pyramid of which the triangle EBC is the base and the point D the vertex.
因此以ABD为底、C为顶点的棱锥等于以DEB为底、C为顶点的棱锥。但后者与以EBC为底、D为顶点的棱锥相同，故前者也等于以EBC为底、D为顶点的棱锥。
Again, since FCBE is a parallelogram, and CE is its diameter, the triangle CEF is equal to the triangle CBE. [I. 34] Therefore also the pyramid of which the triangle BCE is the base and the point D the vertex is equal to the pyramid of which the triangle ECF is the base and the point D the vertex. [XII. 5] But the pyramid of which the triangle BCE is the base and the point D the vertex was proved equal to the pyramid of which the triangle ABD is the base and the point C the vertex; therefore also the pyramid of which the triangle CEF is the base and the point D the vertex is equal to the pyramid of which the triangle ABD is the base and the point C the vertex; therefore the prism ABCDEF has been divided into three pyramids equal to one another which have triangular bases.
又因FCBE是平行四边形，CE为其对角线，故三角形CEF等于三角形CBE。所以以BCE为底、D为顶点的棱锥等于以ECF为底、D为顶点的棱锥。
And, since the pyramid of which the triangle ABD is the base and the point C the vertex is the same with the pyramid of which the triangle CAB is the base and the point D the vertex, for they are contained by the same planes, while the pyramid of which the triangle ABD is the base and the point C the vertex was proved to be a third of the prism in which the triangle ABC is the base and DEF its opposite, therefore also the pyramid of which the triangle ABC is the base and the point D the vertex is a third of the prism which has the same base, the triangle ABC, and DEF as its opposite. PORISM.
但以BCE为底、D为顶点的棱锥已证等于以ABD为底、C为顶点的棱锥，故以CEF为底、D为顶点的棱锥也等于它。因此棱柱被分成三个相等的三角底棱锥。推论：以ABC为底、D为顶点的棱锥是原棱柱的三分之一。