PROPOSITION 7.
Any prism which has a triangular base is divided into three pyramids equal to one another which have triangular bases.
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. [
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.
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.
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.
From this it is manifest that any pyramid is a third part of the prism which has the same base with it and equal height. Q. E. D.