Construction of a Regular Pentagon

Introduction
Also explained is why this construction is correct.

The construction
ABCDE is the pentagon.

 fig. 1

The construction involves the following steps:
1. choose the length of line CD, the base of the pentagon
2. construct center M of CD
3. construct line perpendicular to CD and through D
4. draw N, so DN = DM
5. extend line CN
6. draw circle with center N and radius DN, P is intersection with extended line from 5.
7. extend perpendicular bisector of CD
8. draw circle with center C and radius CP, A is intersection with bisector of CD
9. draw circles with radius CD and centers A, C and D. Points B and E are other angles of pentagon.

Why is this correct?
Look at figure 2:
 fig. 2

Each angle of the pentagon is 108 degrees.
The pentagon may be dissected into 5 equal triangles (with 1 angle in the center)
All angles of the triangles together are 5 * 180 = 900 degrees.
Subtract the angles in the center: 900 - 360 = 540 graden.
Each angle of the pentagon is 540 / 5 = 108 degrees.
Triangle CDE is isosceles, so
LECD = LCED so
LECD = (180 - 108) / 2 = 36 degrees.

Each angle marked + 36 degrees.

CS is bisector of LACD

LCSD = LSDC = 72 degrees, so
CD = CS = SA
Say the length of CD = 1.

We apply this lemma:
In a triangle, the bisector of an angle divides the opposite side in parts
having the same ratio as the sides of the angle
So:
AC : CD = AS : SD
If AC = x we get :
x : 1 = 1 : ( x - 1 ) ...........or
x2 - x - 1 = 0
the ABC rule:
x =  1 2
+  1 2
 \ 5

If we can prove that CP (see figure 1) is equal to
 1 2
+
 1 2

 \ 5

than the construction is correct.

Calculation is left to the reader.

 OneStat