Tetrahedra are Equal iff Bases are Reciprocally Proportional to Heights

Proof

 * Euclid-XII-9.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 equal in volume.

It is to be demonstrated that the areas of their bases is reciprocally proportional to their heights.

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

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

From:

and

it follows that:
 * $BGML = 6 \cdot ABCG$

and:
 * $EHQP = 6 \cdot DEFH$

Therefore:
 * $BGML = EHQP$

But from :
 * $BM : EQ = h \left({EHQP}\right) : h \left({BGML}\right)$

where:
 * $BM$ and $EQ$ are the bases of $BGML$ and $EHQP$ respectively
 * $h \left({EHQP}\right)$ denotes the height of $EHQP$
 * $h \left({BGML}\right)$ denotes the height of $BGML$.

But from :
 * $BM : EQ = \triangle ABC : \triangle DEF$

Therefore by :
 * $\triangle ABC : \triangle DEF = h \left({EHQP}\right) : h \left({BGML}\right)$

But:
 * $h \left({EHQP}\right) = h \left({DEFH}\right)$
 * $h \left({BGML}\right) = h \left({ABCG}\right)$

where:
 * $h \left({ABCG}\right)$ denotes the height of $ABCG$
 * $h \left({DEFH}\right)$ denotes the height of $DEFH$.

Therefore:
 * $\triangle ABC : \triangle DEF = h \left({DEFH}\right) : h \left({ABCG}\right)$

That is the bases of $ABCG$ and $DEFH$ are reciprocally proportional to their heights.

Let $ABCG$ and $DEFH$ be tetrahedra whose bases are reciprocally proportional to their heights:


 * $\triangle ABC : \triangle DEF = h \left({DEFH}\right) : h \left({ABCG}\right)$

It is to be shown that $ABCG$ is equal to $DEFH$.

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

We have that:
 * $\triangle ABC : \triangle DEF = h \left({DEFH}\right) : h \left({ABCG}\right)$

while:
 * $\triangle ABC : \triangle DEF = BM : EQ$

where $BM$ and $EQ$ are the bases of $BGML$ and $EHQP$ respectively.

Therefore by :
 * $BM : EQ = h \left({EHQP}\right) : h \left({BGML}\right)$

From :
 * $BGML = EHQP$

But from:

and

it follows that:
 * $BGML = 6 \cdot ABCG$

and:
 * $EHQP = 6 \cdot DEFH$

Therefore:
 * $ABCG = DEFH$