---
book: 12
number: 7
id: "XII.7"
kind: "theorem"
uses: ["[[book-1/proposition-34]]", "[[book-12/proposition-5]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:12.prop.7"
license: "CC-BY-SA-4.0"
---

# XII.7

*Any prism which has a triangular base is divided into three pyramids equal to one another which have triangular bases.*

## Proof

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*; [[book-1/proposition-34|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. [[book-12/proposition-5|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*. [[book-1/proposition-34|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. [[book-12/proposition-5|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.