Constructing a Regular Heptagon Using Plemelj's Method

step

1

2

3

4

5

6

O

s 7	AE	AC
0.867767	 Abs[1-0.5t] 2 +0.75Abs[t] 2	0.866025

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

1. Draw a circle

σ

with center

O

and radius

1

.

2. Draw an equilateral triangle

OAB

with

AB

on

σ

.

3. Let

C

be the midpoint of

OB

.

4. Construct a point

D

on

OB

so that

OD=1/3OB

.

5. Construct a point

E

on

DC

so that

∠EAC=1/3∠DAC

.

6. Successively measure out points on

σ

at distance

AE

starting with

A

.

Verification

Suppose

∠CAD=3γ

. Then

tan(3γ)=

CD

AC

=

1

6

/



3

2

=

1

3

3

.

Since

cosγ=AC/AE

,

AE=

AC

cosγ

=



3

2cosγ

=s

7

.