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:


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.


, so





so that


Eliminating gives


which has as its only positive solution (see Details).


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



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





Substitute to get the third-degree equation

with solutions




These solutions also follow from the trigonometric identity


Set and to get


which factors as



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

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