Straight Line Perpendicular to Plane from Point is Unique
Theorem
In the words of Euclid:
- From the same point two straight lines cannot be set up at right angles to the same plane on the same side.
(The Elements: Book $\text{XI}$: Proposition $13$)
Proof
Suppose it were possible to set up two straight lines $AB$ and $AC$ perpendicular to the plane of reference and on the same side.
Let a plane be drawn through $BA$ and $AC$.
From Proposition $3$ of Book $\text{XI} $: Common Section of Two Planes is Straight Line:
- let the common section be the straight line $DAE$.
Therefore the straight lines $AB$, $AC$ and $DAE$ are all in the same plane.
We have that $AC$ is perpendicular to the plane of reference.
So from Book $\text{XI}$ Definition $3$: Line at Right Angles to Plane:
- $AC$ is perpendicular to all the straight lines which meet it and are in the plane of reference.
But $DAE$ meets $AC$ and is in the plane of reference.
Therefore $\angle CAE$ is a right angle.
For the same reason, $\angle BAE$ is a right angle.
Therefore $\angle BAE = \angle CAE$.
But both are in the same plane, which is impossible.
Hence the result.
$\blacksquare$
Historical Note
This proof is Proposition $13$ 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