Construction of Straight Line Perpendicular to Plane from point on Plane

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

From :
 * 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 :
 * $AD$ is also perpendicular to the plane of reference.

Therefore $AD$ is the required perpendicular.