Construction of Straight Line Perpendicular to Plane from point not on Plane

Proof

 * Euclid-XI-11.png

Let $A$ be the given elevated point.

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

It is required that a straight line be drawn perpendicular to the plane of reference.

Let any arbitrary straight line $BC$ be drawn in the plane of reference.

From :
 * Let $AD$ be drawn from $A$ perpendicular to $BC$.

If $AD$ is perpendicular to the plane of reference, then the construction is complete.

Otherwise, let $DE$ be drawn from $D$ perpendicular to $BC$ and in the plane of reference.

From :
 * Let $AF$ be drawn from $A$ perpendicular to $DE$.

From :
 * Let $GH$ be drawn from the point $F$ parallel to $BC$.

We have that $BC$ is perpendicular to each of $DA$ and $DE$.

Therefore from :
 * $BC$ is perpendicular to the plane through $ED$ and $DA$.

We have that $GH$ is parallel to $BC$.

But from :
 * $GH$ is also perpendicular to the plane through $ED$ and $DA$.

Therefore by :
 * $GH$ is perpendicular to all the straight lines which meet it and are in the plane through $ED$ and $DA$.

But $AF$ meets $GH$ and is in the plane through $ED$ and $DA$.

Therefore $GH$ is perpendicular to $FA$.

So $FA$ is perpendicular to $GH$.

But $FA$ is also perpendicular to $DE$.

Therefore $FA$ is perpendicular to both $GH$ and $DE$.

Therefore from :
 * $AF$ is perpendicular to the plane through $DE$ and $GH$.

But the plane through $DE$ and $GH$ is the plane of reference.

Therefore $AF$ is perpendicular to the plane of reference.

Thus $AF$ is the required line.