# 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