Description of the infinite family of pentagons:

1. Three of the five vertices of the pentagons are always
the same (independently of the value of D). They are the three vertices
of an isosceles triangle whose base angles are 72^{o}. This triangle
is pictured here. Two of the vertices are the
endpoints of the base. They will be denoted by v_{1}
and v_{2}. The apex of this isosceles triangle
will be another vertex of the pentagon, which will be denoted by v_{3}.
An equivalent way to describe this isosceles triangle is to require that
the ratio **d**/**b**
is equal to the Golden Section *phi *= (1+\/5)/2
. The equivalence results from the fact that cos(72^{o})
= 1/2*phi.*

2. Suppose that D is an angle between 36^{o}
and 180^{o}. On the same base (of length b), construct another
isosceles triangle whose apex is within the previous triangle and such
that the angle at that apex is D. The apex of this new, smaller triangle
will be referred to as the "*special interior point*." Here
is a picture. The new triangle is in red, the
*special
interior point* is called w. The base angles of this
new triangle are equal to (180^{o} - D)/2
.

3. We now describe how to get the remaining two
vertices of the pentagon. Consider the line segment joining w and v_{3}.
Using this line segment as a base, construct two equilateral triangles.
Here is a picture . The
two equilateral triangles are in purple. The remaining two vertices of
the pentagon which corresponds to a given angle D will be called v_{4}
and v_{5}. They are the indicated vertices
of the two equilateral triangles.

4. The five vertices v_{1},
. . . , v_{5} are the vertices of the
pentagon.