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
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.
- 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.
- $AD$ 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
- 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 3 (2nd ed.) ... (previous) ... (next): Book $\text{XI}$. Propositions