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).