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 $OE=CECO=CDCA=AD=AB$.
If you are interested in seeing the rules for compass and straightedge constructions, click on the provided.
References
 1 E. J. Nyström: Korkeamman geometrian alkeet sovellutuksineen. Kustannusosakeyhtiö Otava, Helsinki (1948).
Title  circle with given center and given radius 

Canonical name  CircleWithGivenCenterAndGivenRadius 
Date of creation  20130322 17:14:03 
Last modified on  20130322 17:14:03 
Owner  Wkbj79 (1863) 
Last modified by  Wkbj79 (1863) 
Numerical id  30 
Author  Wkbj79 (1863) 
Entry type  Algorithm 
Classification  msc 51M15 
Classification  msc 5100 