Constructing a Regular Heptagon Using Plemelj's Method
This Demonstration shows Plemelj's method for constructing a regular heptagon, using the following steps:[more]
1. Draw a circle with center and radius .
2. Draw an equilateral triangle with on .
3. Let be the midpoint of .
4. Construct a point on so that .
5. Construct a point on so that .
6. Successively measure out points on at distance starting with .
Suppose . Then
This method for constructing a regular heptagon using angle trisection was found by Plemelj in 1892 and published in 1912. The construction is taken from [1, pp. 183–184]. The approximation for was known to Abûl-Wefâ and Heron of Alexandria [1, p. 184].
Start with the trigonometric identity
replace by and by in the trigonometric identity to get the cubic
Since and , take .
Substitute to get
Substitute to get the Vieta form of the equation,
to get the positive solutions
 G. E. Martin, Geometric Constructions, New York: Springer, 1998.