Any prism with a triangular base is divided into three pyramids equal to one another with triangular bases.

Let there be a prism with the triangular base *ABC* and *DEF* opposite.

I say that the prism *ABCDEF* is divided into three pyramids equal to one another, which have triangular bases.

Join *BD, EC,* and *CD.*

Since *ABED* is a parallelogram, and *BD* is its diameter, therefore the triangle *ABD* equals the triangle *EBD.* Therefore the pyramid with triangular base *ABD* and vertex *C* equals the pyramid with triangular base *DEB* and vertex *C.*

But the pyramid with triangular base *DEB* and vertex *C* is identical with the pyramid with triangular base *EBC* and vertex *D,* for they are contained by the same planes.

Therefore the pyramid with triangular base *ABD* vertex *C* is also equal to the pyramid with triangular base *EBC* and vertex *D.*

Again, since *FCBE* is a parallelogram, and *CE* is its diameter, therefore the triangle *CEF* equals the triangle *CBE.*

Therefore the pyramid with triangular base *BCE* and vertex *D* equals the pyramid with triangular base *ECF* and vertex *D.*

But the pyramid with triangular base *BCE* and vertex *D* was proved equal to the pyramid with triangular base *ABD* and vertex *C,* therefore the pyramid with triangular base *CEF* and vertex *D* equals the pyramid with triangular base *ABD* and vertex *C.* Therefore the prism *ABCDEF* is divided into three pyramids equal to one another which have triangular bases.

And, since the pyramid with triangular base *ABD* and vertex *C* is identical with the pyramid with triangular base *CAB* and vertex *D,* for they are contained by the same planes, while the pyramid with triangular base *ABD* vertex *C* was proved to be a third of the prism with triangular base *ABC* and *DEF* opposite, therefore the pyramid with triangular base *ABC* and vertex *D* is a third of the prism with the same base *ABC,* and *DEF* opposite.

Therefore, *any prism with a triangular base is divided into three pyramids equal to one another with triangular bases.*

Q.E.D.

From this it is clear that any pyramid is a third part of the prism with the same base and equal height.