Proof: By Euclid
(related to Proposition: Prop. 11.01: Straight Line cannot be in Two Planes)
- For, if possible, let some part, $AB$, of the straight line $ABC$ be in a reference plane, and some part, $BC$, in a more elevated (plane).
- In the reference plane, there will be some straight line continuous with, and straight-on to, $AB$.
- Let it be $BD$.
- Thus, $AB$ is a common segment of the two (different) straight lines $ABC$ and $ABD$.
- The very thing is impossible, inasmuch as if we draw a circle with center $B$ and radius $AB$ then the diameters ($ABD$ and $ABC$) will cut off unequal circumferences of the circle.
- Thus, some part of a straight line cannot be in a reference plane, and (some part) in a more elevated (plane).
- (Which is) the very thing it was required to show.
Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"