Given a line $AB$ and a point $C$ not on $AB$ it is easy enough to construct a perpendicular line to $AB$ passing through $C$ using two circles as demonstrated in the following picture.
Here we pick two arbitrary points $E$ and $F$ on $AB$ and draw circles with centres $E$ and $F$ and radii $EC$ and $FC$. Then we take the two points of intersection of these circles (one of which is $C$) and draw a line between them. This gives a perpendicular line to $AB$ passing through $C$.
A similar proof is given in Euclid's Elements which uses the same idea and constructs two circles.
I am interested to know if we can construct a perpendicular from a given point to given line using the compass just a single time.