Straight Line cannot be in Two Planes

Theorem

In the words of Euclid:

A part of a straight line cannot be in the plane of reference and part in a plane more elevated.

(The Elements: Book $\text{XI}$: Proposition $1$)

Proof

Suppose it were possible to have a straight line in more than one plane.

Let a part $AB$ of the straight line $ABC$ be in the plane of reference, and another part $BC$ be in a plane more elevated.

There will then be in the plane of reference some straight line $BD$ continuous with $AB$ in a straight line.

Therefore $AB$ is a common segment of the two straight lines $ABC$ and $ABD$.

Suppose a circle is described with center $B$ and radius $AB$.

Then the diameters would cut off unequal arcs of the circle.

Hence the result.

$\blacksquare$

Historical Note

This proof is Proposition $1$ 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