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 72o. This triangle is pictured here. Two of the vertices are the endpoints of the base. They will be denoted by v1 and v2. The apex of this isosceles triangle will be another vertex of the pentagon, which will be denoted by v3. 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(72o) = 1/2phi.
2. Suppose that D is an angle between 36o
and 180o. 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 (180o - D)/2
.
3. We now describe how to get the remaining two vertices of the pentagon. Consider the line segment joining w and v3. 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 v4 and v5. They are the indicated vertices of the two equilateral triangles.
4. The five vertices v1,
. . . , v5 are the vertices of the
pentagon.
