PROPOSITION 9.
In equal pyramids which have triangular bases the bases are reciprocally proportional to the heights; and those pyramids in which the bases are reciprocally proportional to the heights are equal.
For let there be equal pyramids which have the triangular bases
ABC,
DEF and vertices the points
G,
H; I say that in the pyramids
ABCG,
DEFH the bases are reciprocally proportional to the heights, that is, as the base
ABC is to the base
DEF, so is the height of the pyramid
DEFH to the height of the pyramid
ABCG.
For let the parallelepipedal solids
BGML,
EHQP be completed.
Now, since the pyramid
ABCG is equal to the pyramid
DEFH, and the solid
BGML is six times the pyramid
ABCG, and the solid
EHQP six times the pyramid
DEFH, therefore the solid
BGML is equal to the solid
EHQP.
But in equal parallelepipedal solids the bases are reciprocally proportional to the heights; [
XI. 34] therefore, as the base
BM is to the base
EQ, so is the height of the solid
EHQP to the height of the solid
BGML.
But, as the base
BM is to
EQ, so is the triangle
ABC to the triangle
DEF. [
I. 34]
Therefore also, as the triangle
ABC is to the triangle
DEF, so is the height of the solid
EHQP to the height of the solid
BGML. [
V. 11]
But the height of the solid
EHQP is the same with the height of the pyramid
DEFH, and the height of the solid
BGML is the same with the height of the pyramid
ABCG, therefore, as the base
ABC is to the base
DEF, so is the height of the pyramid
DEFH to the height of the pyramid
ABCG.
Therefore in the pyramids
ABCG,
DEFH the bases are reciprocally proportional to the heights.
Next, in the pyramids
ABCG,
DEFH let the bases be reciprocally proportional to the heights; that is, as the base
ABC is to the base
DEF, so let the height of the pyramid
DEFH be to the height of the pyramid
ABCG; I say that the pyramid
ABCG is equal to the pyramid
DEFH.
For, with the same construction, since, as the base
ABC is to the base
DEF, so is the height of the pyramid
DEFH to the height of the pyramid
ABCG, while, as the base
ABC is to the base
DEF, so is the parallelogram
BM to the parallelogram
EQ, therefore also, as the parallelogram
BM is to the parallelogram
EQ, so is the height of the pyramid
DEFH to the height of the pyramid
ABCG. [
V. 11]
But the height of the pyramid
DEFH is the same with the height of the parallelepiped
EHQP, and the height of the pyramid
ABCG is the same with the height of the parallelepiped
BGML; therefore, as the base
BM is to the base
EQ, so is the height of the parallelepiped
EHQP to the height of the parallelepiped
BGML.
But those parallelepipedal solids in which the bases are reciprocally proportional to the heights are equal; [
XI. 34] therefore the parallelepipedal solid
BGML is equal to the parallelepipedal solid
EHQP.
And the pyramid
ABCG is a sixth part of
BGML, and the pyramid
DEFH a sixth part of the parallelepiped
EHQP; therefore the pyramid
ABCG is equal to the pyramid
DEFH.
Therefore etc Q. E. D.