Common Sections of Parallel Planes with other Plane are Parallel
Theorem
In the words of Euclid:
- If two parallel planes be cut by any plane, their common sections are parallel.
(The Elements: Book $\text{XI}$: Proposition $16$)
Proof
Let $AB$ and $CD$ be two planes which are parallel.
Let $AB$ and $CD$ be cut by the plane $EFGH$.
Let:
- $EF$ be the common section of $EFGH$ and $AB$
and:
- $GH$ be the common section of $EFGH$ and $CD$.
It is to be shown that $EF$ is parallel to $GH$.
Suppose that $EF$ and $GH$ are not parallel.
Then when produced, they will meet, either in the direction of $F$ and $H$, or in the direction of $E$ and $G$.
WLOG suppose they meet at $K$ when produced in the direction of $F$ and $H$.
We have that $EFK$ is in the plane $AB$.
Therefore from Proposition $1$ of Book $\text{XI} $: Straight Line cannot be in Two Planes:
But $K$ is one of the points of $EFK$.
Therefore $K$ is in the plane $AB$.
For the same reason, $K$ is also in the plane $CD$.
Therefore the planes $AB$ and $CD$ will meet when produced.
But they do not meet, because they are parallel.
Therefore $EF$ and $GH$ will not meet when produced in the direction of $F$ and $H$.
By the same argument, $EF$ and $GH$ will not meet when produced in the direction of $E$ and $G$.
But from Book $\text{I}$ Definition $23$: Parallel Lines:
- $EF$ is parallel to $GH$.
$\blacksquare$
Historical Note
This proof is Proposition $16$ 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