Constructing a Regular Heptagon Using Gleason's Method

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This Demonstration shows Gleason's method for constructing a regular heptagon, using the following steps:

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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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Contributed by: Izidor Hafner (September 2017)
Open content licensed under CC BY-NC-SA


Snapshots


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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