G.558 prism triangular base — Heath

XII.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 < A B C > is the base and < D E F > its opposite; I say that the prism < A B C D E F > is divided into three pyramids equal to one another, which have triangular bases.

For let / B D /, / E C /, / C D / be joined.

Since < A B E D > is a parallelogram, and / B D / is its diameter,

therefore the triangle < A B D > is equal to the triangle < E B D >; [G.62] therefore also the pyramid of which the triangle < A B D > is the base and the point C the vertex is equal to the pyramid of which the triangle < D E B > is the base and the point C the vertex. [G.556]

But the pyramid of which the triangle < D E B > is the base and the point C the vertex is the same with the pyramid of which the triangle < E B C > is the base and the point D the vertex; for they are contained by the same planes.

Therefore the pyramid of which the triangle < A B D > is the base and the point C the vertex is also equal to the pyramid of which the triangle < E B C > is the base and the point D the vertex.

Again, since < F C B E > is a parallelogram, and / C E / is its diameter,

[G.62]the triangle < C E F > is equal to the triangle < C B E >.

Therefore also the pyramid of which the triangle < B C E > is the base and the point D the vertex is equal to the pyramid of which the triangle < E C F > is the base and the point D the vertex. [G.556]

But the pyramid of which the triangle < B C E > is the base and the point D the vertex was proved equal to the pyramid of which the triangle < A B D > is the base and the point C the vertex;

therefore also the pyramid of which the triangle < C E F > is the base and the point D the vertex is equal to the pyramid of which the triangle < A B D > is the base and the point C the vertex; therefore the prism < A B C D E F > has been divided into three pyramids equal to one another which have triangular bases.

And, since the pyramid of which the triangle < A B D > is the base and the point C the vertex is the same with the pyramid of which the triangle < C A B > is the base and the point D the vertex, for they are contained by the same planes, while the pyramid of which the triangle < A B D > is the base and the point C the vertex was proved to be a third of the prism in which the triangle < A B C > is the base and < D E F > its opposite,

therefore also the pyramid of which the triangle < A B C > is the base and the point D the vertex is a third of the prism which has the same base, the triangle < A B C >, and < D E F > as its opposite.

PORISM. From this it is manifest that any pyramid is a third :term:`part of <G.253>` the prism which has the same base with it and equal :term:`height <G.209>`. Q. E. D.

Dependency Graph

Required for

G.559, G.563, G.565, G.570, G.571, G.572, G.573, G.577, G.578