Construction of Straight Line Perpendicular to Plane from point on Plane

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Theorem

In the words of Euclid:

To set up a straight line at right angles to a given plane from a given point on it.

(The Elements: Book $\text{XI}$: Proposition $12$)


Proof

Euclid-XI-12.png

Let the given plane be identified as the plane of reference.

Let $A$ be the point from which the perpendicular is to be constructed.

Let $B$ be any elevated point.

From Proposition $11$ of Book $\text{XI} $: Construction of Straight Line Perpendicular to Plane from point not on Plane:

let $BC$ be constructed from $B$ perpendicular to the plane of reference.

From Proposition $31$ of Book $\text{I} $: Construction of Parallel Line:

let $AD$ be drawn from $A$ parallel to $BC$.

We have that $AD$ and $BC$ are two parallel straight lines of which one of them $BC$ is perpendicular to the plane of reference.

From Proposition $8$ of Book $\text{XI} $: Line Parallel to Perpendicular Line to Plane is Perpendicular to Same Plane:

$AD$ is also perpendicular to the plane of reference.

Therefore $AD$ is the required perpendicular.

$\blacksquare$


Historical Note

This proof is Proposition $12$ of Book $\text{XI}$ of Euclid's The Elements.


Sources