Common Section of Two Planes is Straight Line

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