Constructing a Regular Heptagon Using Plemelj's Method

This Demonstration shows Plemelj's method for constructing a regular heptagon, using the following steps:
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 .
Verification
Suppose . Then
.
Since ,
.

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DETAILS

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
.
Since
,
replace by and by in the trigonometric identity to get the cubic
.
Then
.
Since and , take .
Substitute to get
.
Substitute to get the Vieta form of the equation,
.
Set
to get the positive solutions
,
,
.
So
,
where
.
Then
.
Reference
[1] G. E. Martin, Geometric Constructions, New York: Springer, 1998.
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