Common Section of Two Planes is Straight Line
Jump to navigation
Jump to search
Theorem
In the words of Euclid:
- If two planes cut one another, their common section is a straight line.
(The Elements: Book $\text{XI}$: Proposition $3$)
Proof
Let $AB$ and $BC$ be two distinct planes that cut one another.
Let $DB$ be their common section.
Suppose $DB$ were not a straight line.
Then let:
- the straight line segment $DEB$ be drawn in the planes $AB$
and:
- the straight line segment $DFB$ be drawn in the planes $BC$.
Thus the two straight line segments $DEB$ and $DFB$ have the same endpoints.
Thus $DEB$ and $DFB$ enclose an area, which is absurd.
Therefore $DEB$ and $DFB$ are not straight lines.
Hence the result.
$\blacksquare$
Historical Note
This proof is Proposition $3$ 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