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[parent] construction of regular $2n$-gon from regular $n$-gon (Algorithm)

Given a regular $ n$-gon, one can construct a regular $ 2n$-gon using compass and straightedge. This procedure will be demonstrated by starting with a regular pentagon; the procedure will thus produce a regular decagon.

The procedure is as follows:

  1. Bisect two of the interior angles of the regular polygon. These angle bisectors will intersect at the center of the regular polygon.

    \begin{pspicture}(-4,-3)(4,4) \pspolygon(0,3.404)(-3.238,1.052)(-2,-2.754)(2,-2.... ...ts(0,3.404)(-3.238,1.052)(-2,-2.754)(2,-2.754)(3.238,1.052)(0,0) \end{pspicture}
  2. Connect each vertex of the regular polygon to the center.

    \begin{pspicture}(-4,-3)(4,4) \pspolygon(0,3.404)(-3.238,1.052)(-2,-2.754)(2,-2.... ...ts(0,3.404)(-3.238,1.052)(-2,-2.754)(2,-2.754)(3.238,1.052)(0,0) \end{pspicture}
  3. Construct the circumscribed circle of the regular polygon.

    \begin{pspicture}(-4,-4)(4,4) \pspolygon(0,3.404)(-3.238,1.052)(-2,-2.754)(2,-2.... ...ts(0,3.404)(-3.238,1.052)(-2,-2.754)(2,-2.754)(3.238,1.052)(0,0) \end{pspicture}
  4. Bisect each of the central angles of the circle to obtain the points where the angle bisectors intersect the circle.

    \begin{pspicture}(-4,-4)(4,4) \pspolygon(0,3.404)(-3.238,1.052)(-2,-2.754)(2,-2.... ...)(0,0)(0,-3.404)(-3.238,-1.052)(-2,2.754)(2,2.754)(3.238,-1.052) \end{pspicture}
  5. Connect the dots to form the regular $ 2n$-gon. In the picture below, all drawn figures except for the original polygon, the circle, and the formed polygon are drawn in cyan to emphasize the three figures that are not dashed.

    \begin{pspicture}(-4,-4)(4,4) \psarc[linecolor=cyan](0,3.404){0.7}{195}{345} \ps... ...)(0,0)(0,-3.404)(-3.238,-1.052)(-2,2.754)(2,2.754)(3.238,-1.052) \end{pspicture}

This construction is justified because the triangles formed by the drawn radii of the circle and the drawn (blue) polygon are congruent by SAS (note that all of the central angles have measure $ \frac{360^{\circ}}{2n}$), giving that all of the sides and all of the interior angles of the drawn polygon are congruent.

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



"construction of regular $2n$-gon from regular $n$-gon" is owned by Wkbj79.
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Cross-references: compass and straightedge constructions, sides, SAS, congruent, radii, triangles, polygon, points, central angles, circle, circumscribed, vertex, intersect, angle bisectors, regular polygon, interior angles, regular decagon, pentagon, straightedge, compass
There are 2 references to this entry.

This is version 16 of construction of regular $2n$-gon from regular $n$-gon, born on 2007-06-25, modified 2007-10-08.
Object id is 9677, canonical name is ConstructionOfRegular2nGonFromRegularNGon.
Accessed 703 times total.

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