 # 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

 G. E. Martin, Geometric Constructions, New York: Springer, 1998 p. 45.

## Permanent Citation

Izidor Hafner

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