Proof: By Euclid
(related to Proposition: Prop. 11.13: Straight Line Perpendicular to Plane from Point is Unique)
- So it will make a straight cutting (passing) through (point) $A$ in the reference plane [Prop. 11.3].
- Let it have made $DAE$.
- Thus, $AB$, $AC$, and $DAE$ are straight lines in one plane.
- And since $CA$ is at right angles to the reference plane, it will thus also make right angles with all of the straight lines joined to it which are also in the reference plane [Def. 11.3] .
- And $DAE$, which is in the reference plane, is joined to it.
- Thus, angle $CAE$ is a right angle.
- So, for the same (reasons), $BAE$ is also a right angle.
- Thus, $CAE$ (is) equal to $BAE$.
- And they are in one plane.
- The very thing is impossible.
- Thus, two (different) straight lines cannot be set up at the same point at right angles to the same plane, on the same side.
- (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"