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]