4. In Project 3 we can see that there are many similar triangles in the pentagram.
both ΔAFI and ΔABI are 36-72-72 triangles, so they are similar. If we let AI represent 1 unit of length, then since ΔAFI is isosceles, BF is also 1 unit in length. Since ΔABF is also isosceles, AF is also 1 unit in length. Let x denote the length of FG. Then since
ΔAFI ∼ ΔBAI
we get the following ratio and proportion
which is the definition of the Proportion of the Golden Mean.
To solve for x, clear denominators
And use the quadratic formula
If we take the negative square root, we will get a negative answer, and since x is a distance, and distances are never negative, we must mean
Other instances of the proportion of the golden mean in the pentagram include
The triangles fall into two similarity classes: the 36-72-72 triangles and the 36-36-108 triangles.
ΔAFJ, ΔABG, and ΔACD are 36-72-72 triangles.
ΔABF and ΔABC are 36-36-108 triangles.
Both types are called Golden Triangles. There are thirty golden triangles in a pentagram.