Perpendicular to a Point NOT on a Line

How to construct a Perpendicular to a Point NOT on a Line
using just a compass and a straightedge

Steps to Construct

Here are the steps we follow in the animation:

1. Place the compass point on the given point P
2. Open the compass wide enough to cross the line in two places. Draw an arc that cuts the line at two points, let's call them A and B
3. Place the compass point on A and draw an arc on the opposite side of the line from P
4. Without changing the compass width, place the compass point on B and draw another arc that crosses the one we just drew. Let's call this intersection point Q
5. Use a straightedge to draw a line through points P and Q

The line PQ is perpendicular to our original line and passes through P!