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Constructing a Regular Heptagon Using Plemelj's Method

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step
1
2
3
4
5
6
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=1/3OB
.
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
.
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