Constructing a Regular Heptagon Using Lill's Method

This Demonstration shows how to construct a regular heptagon using Lill's method for solving cubic equations.
The points of a regular heptagon with vertices on a circle of radius 1 are given by . Since is a solution, if we divide the polynomial by , we get
.
If , then
.
Substituting
leads to the cubic equation
.
It has solutions , , .
This follows from the trigonometric identity
.
Set and to get
,
which factors as
.
There are solutions when the points and coincide.

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Reference
[1] G. E. Martin, Geometric Constructions, New York: Springer, 1998 p. 45.
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