circle with given center and given radius

Task.  Draw the circle having a given point $O$ as its center and a given line segment of length $AB$ as its radius. This construction must be performed with constraints in the spirit of Euclid: One must not take the length of $\overline{AB}$ between the tips of the compass (i.e. (http://planetmath.org/Ie), one must pretend that the compass is collapsible (http://planetmath.org/CollapsibleCompass)). This means than one may only draw arcs that are of circles with the center and one point of the circumference known.

Solution.

1. Draw an arc of the circle $a$ through $A$ with center $O$ and an arc of the circle $o$ through $O$ with center $A$. These arcs must intersect each other. Let one of the intersection points be $C$.

2. Draw the lines $\overleftrightarrow{CA}$ and $\overleftrightarrow{CO}$.

3. Draw an arc of the circle $b$ through $B$ and with center $A$. Let $D$ be the intersection point of $b$ and the line $\overleftrightarrow{CA}$.

4. Draw an arc of the circle $c$ through $C$ and with center $D$. Let $E$ be the intersection point of $d$ and the line $\overleftrightarrow{CO}$ with $E\ne C$.

5. Draw the circle $e$ through $E$ and with center $O$. This is the required circle.

A justification for this construction is that $OE=CE-CO=CD-CA=AD=AB$.

If you are interested in seeing the rules for compass and straightedge constructions, click on the provided.