Kitap 12
6
And, since ADEM, FGHN are two pyramids which have triangular bases and equal height, therefore, as the base ADE is to the base FGH, so is the pyramid ADEM to the pyramid FGHN. [XII. 5]
6
But, as the base ADE is to the base ABCDE, so was the pyramid ADEM to the pyramid ABCDEM.
6
Therefore also, ex aequali, as the base ABCDE is to the base FGH, so is the pyramid ABCDEM to the pyramid FGHN. [V. 22]
6
But further, as the base FGH is to the base FGHKL, so also was the pyramid FGHN to the pyramid FGHKLN.
6
Therefore also, ex aequali, as the base ABCDE is to the base FGHKL, so is the pyramid ABCDEM to the pyramid FGHKLN. [V. 22] Q. E. D.
PROPOSITION 7.
7
Any prism which has a triangular base is divided into three pyramids equal to one another which have triangular bases.
7
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.
7
For let BD, EC, CD be joined.
7
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. [XII. 5]
7
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.
7
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.
7
Again, since FCBE is a parallelogram, and CE is its diameter, the triangle CEF is equal to the triangle CBE. [I. 34]
7
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]
