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[parent] circle with given center and given radius (Algorithm)

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., one must pretend that the compass is collapsible). This means than one may only draw arcs that are of circles with the center and one point of the circumference known.


\begin{pspicture}(-2,-3)(4,1) \psline(-2,-2)(3,-2) \psdots(-2,-2)(3,-2)(0,1) \rp... ...$} \rput[a](3.3,-2.3){$B$} \rput[a](0,0.7){$O$} \rput[b](0,1){.} \end{pspicture}

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$.

    \begin{pspicture}(-4,-4)(4,2) \psline(-2,-2)(3,-2) \psarc[linecolor=blue](0,1){3... ...put[a](3.3,-2.3){$B$} \rput[a](0,0.7){$O$} \rput[a](-4,1.5){$C$} \end{pspicture}
  2. Draw the lines $ \overleftrightarrow{CA}$ and $ \overleftrightarrow{CO}$.

    \begin{pspicture}(-6,-6)(6,6) \psline(-2,-2)(3,-2) \psarc(0,1){3.60555}{150}{330... ...put[a](3.3,-2.3){$B$} \rput[a](0,0.7){$O$} \rput[a](-4,1.5){$C$} \end{pspicture}
  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}$.

    \begin{pspicture}(-6,-7)(6,6) \psline(-2,-2)(3,-2) \psarc(0,1){3.60555}{150}{330... ...put[a](0,0.7){$O$} \rput[a](-4,1.5){$C$} \rput[l](0.4,-6.6){$D$} \end{pspicture}
  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 \neq C$.

    \begin{pspicture}(-6,-7)(7,6) \rput[l](-5.869,-0.397){.} \rput[r](6.3,-0.397){.}... ...[a](-4,1.5){$C$} \rput[l](0.4,-6.6){$D$} \rput[a](5.05,0.3){$E$} \end{pspicture}
  5. Draw the circle $ e$ through $ E$ and with center $ O$. This is the required circle.

    \begin{pspicture}(-6,-7)(7,6) \rput[l](-5.869,-0.397){.} \rput[r](6.3,-0.397){.}... ...[a](-4,1.5){$C$} \rput[l](0.4,-6.6){$D$} \rput[a](5.05,0.3){$E$} \end{pspicture}

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 link provided.

Bibliography

1
E. J. NYSTRÖM: Korkeamman geometrian alkeet sovellutuksineen. Kustannusosakeyhtiö Otava, Helsinki (1948).



"circle with given center and given radius" is owned by Wkbj79. [ full author list (2) | owner history (1) ]
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Cross-references: compass and straightedge constructions, lines, intersect, circumference, arcs, compass, radius, length, line segment, center, point, circle
There are 3 references to this entry.

This is version 27 of circle with given center and given radius, born on 2007-06-11, modified 2007-06-13.
Object id is 9563, canonical name is CircleWithGivenCenterAndGivenRadius.
Accessed 948 times total.

Classification:
AMS MSC: 51-00 (Geometry :: General reference works )
 51M15 (Geometry :: Real and complex geometry :: Geometric constructions)
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