Construction of Perpendicular using Rusty Compass

Theorem
Let $AB$ be a line segment.

Using a straightedge and rusty compass, it is possible to construct a straight line at right angles to $AB$ from the endpoint $A$, without extending $AB$ past $A$.

Construction

 * Perpendicular-line-rusty-compass.png

Mark $AC$ equal to the fixed radius of the rusty compass.

Construct two arcs whose centers are at $A$ and $C$ respectively of radius $AC$.

Let them meet at $D$.

Produce $CD$.

Measure $E$ from $D$ using the rusty compass so that $DE = CD$.

Then $AE$ is the required straight line at right angles to $AB$.

Proof
As $DE = CD = DA$, the points $A$, $C$ and $E$ all lie on a circle of radius $AC$.

$CE$ is a straight line through the centers of circle $ACE$ and so is a diameter of circle $ACE$.

Hence by Thales' Theorem, $\angle CAE$ is a right angle