Volumes of Similar Tetrahedra are in Triplicate Ratio of Corresponding Sides

Proof

 * Euclid-XII-8.png

Let $ABCG$ be a tetrahedron whose base is $\triangle ABC$ and whose apex is $G$.

Let $DEFH$ be a tetrahedron whose base is $\triangle DEF$ and whose apex is $H$.

Let $ABCG$ and $DEFH$ be similar and similarly situated.

It is to be demonstrated that the ratio of $ABCG$ to $DEFH$ is triplicate of the ratio of the sides $BC$ and $EF$.

Let the parallelepipeds $BGML$ and $EHQP$ be completed.

We have that $ABCG$ is similar to $DEFH$.

Therefore:
 * $\angle ABC = \angle DEF$
 * $\angle GBC = \angle HEF$
 * $\angle ABG = \angle DEH$

and:
 * $AB : DE = BC : EF = BG : EH$

It follows that the parallelogram $BM$ is similar to the parallelogram $EQ$.

For the same reason:
 * the parallelogram $BN$ is similar to the parallelogram $ER$

and:
 * the parallelogram $BK$ is similar to the parallelogram $EO$

We also have that:
 * $BM, BK, BN$ are similar to $EQ, EO, ER$.

But by :
 * the three parallelograms $BM, BK, BN$ are equal and similar to their opposites

and
 * the three parallelograms $EQ, EO, ER$ are equal and similar to their opposites.

Therefore the parallelepipeds $BGML$ and $EHQP$ are contained by equal numbers of similar planes.

Therefore $BGML$ is similar to $EHQP$.

So by :
 * $BGML : EQHP = BC^3 : EF^3$

But from:

and

it follows that:
 * $BGML : EQHP = ABCG : DEFH$

Hence the result.