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Constructing a Regular Heptagon Using Gleason's Method

This Demonstration shows Gleason's method for constructing a regular heptagon, using the following steps:
1. Draw a line segment of length 2 with midpoint and a circle with center and radius . Let , and be points on the line segment such that and . Draw an equilateral triangle .
2. Draw a point between and so that . Draw an arc with center and radius . Let . The ray through with angle to meets the arc at a point .
3. The line perpendicular to through meets at and meets the circle at .
4. The side length of the heptagon is and a compass can be used to measure out the other vertices of the heptagon.
Verification
, so
.
Therefore
.
Define
so that
.
Eliminating gives
,
which has as its only positive solution (see Details).

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DETAILS

The points of a regular heptagon inscribed in the circle of radius 1 are given by . Since is a solution, divide the polynomial by to get
.
If
then
.
Substitute to get the third-degree equation
with solutions
,
,
.
These solutions also follow from the trigonometric identity
.
Set and to get
,
which factors as
.
Reference
[1] G. E. Martin, Geometric Constructions, New York: Springer, 1998 p. 45.
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