construction of regular $2n$gon from regular $n$gon
Given a regular^{} $n$gon (http://planetmath.org/RegularPolygon), 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 (http://planetmath.org/Center9) of the regular polygon.

2.
Connect each vertex of the regular polygon to the center.

3.
Construct the circumscribed circle of the regular polygon.

4.
Bisect each of the central angles^{} of the circle to obtain the points where the angle bisectors intersect the circle.

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.
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 (http://planetmath.org/AngleMeasure) $\frac{{360}^{\circ}}{2n}$), 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 provided.
Title  construction of regular $2n$gon from regular $n$gon 

Canonical name  ConstructionOfRegular2ngonFromRegularNgon 
Date of creation  20130322 17:19:32 
Last modified on  20130322 17:19:32 
Owner  Wkbj79 (1863) 
Last modified by  Wkbj79 (1863) 
Numerical id  19 
Author  Wkbj79 (1863) 
Entry type  Algorithm 
Classification  msc 51M15 
Classification  msc 5100 