Opposite Planes of Solid contained by Parallel Planes are Equal Parallelograms
Theorem
In the words of Euclid:
- If a solid be contained by parallel planes, the opposite planes in it are equal and parallelogrammic.
(The Elements: Book $\text{XI}$: Proposition $24$)
Proof
Let the solid $CDHG$ be contained by the parallel planes $AC, GF, AH, DF, BF, AE$.
It is to be demonstrated that opposite planes are equal parallelograms.
We have that the two parallel planes $BG$ and $CE$ are cut by the plane $AC$.
- the common sections of $BG$ and $CE$ are parallel lines.
Thus $AB \parallel DC$.
Again, we have that the two parallel planes $BF$ and $AE$ are cut by the plane $AC$.
Thus $BC \parallel AD$.
But $AB \parallel DC$.
Therefore $AC$ is by definition a parallelogram.
Similarly it can be shown that each of $GF, AH, DF, BF, AE$ are parallelograms.
Let $AH$ and $DF$ be joined.
We have that:
- $AB \parallel DC$
and:
- $BH \parallel CF$
Thus the two straight lines $AB$ and $BH$ which meet one another are parallel to the two straight lines $DC$ and $CF$ which also meet one another, but not in the same plane.
Therefore by Proposition $10$ of Book $\text{XI} $: Two Lines Meeting which are Parallel to Two Other Lines Meeting contain Equal Angles:
- $\angle ABH = \angle DCF$
From Proposition $34$ of Book $\text{I} $: Opposite Sides and Angles of Parallelogram are Equal:
- $AB$ and $BH$ are equal to $DC$ and $CF$.
and because:
- $\angle ABH = \angle DCF$
it follows that:
- $AH = DF$
So from Proposition $4$ of Book $\text{I} $: Triangle Side-Angle-Side Congruence:
- $\triangle ABH = \triangle DCF$
We have that the parallelogram $BG$ is double $\triangle ABH$.
From Proposition $34$ of Book $\text{I} $: Opposite Sides and Angles of Parallelogram are Equal:
- the parallelogram $CE$ is double the $\triangle DCF$.
Therefore the parallelogram $BG$ equals the parallelogram $CE$.
Similarly it is shown that:
- the parallelogram $AC$ equals the parallelogram $GF$
and:
- the parallelogram $AE$ equals the parallelogram $BF$.
$\blacksquare$
Historical Note
This proof is Proposition $24$ of Book $\text{XI}$ of Euclid's The Elements.
Sources
- 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 3 (2nd ed.) ... (previous) ... (next): Book $\text{XI}$. Propositions