4. In Project 3 we can see that there are many similar triangles in the pentagram.
In particular,
both triangle BFG and triangle AGB are 36-72-72 triangles, so they are similar. If we let BG represent 1 unit of length, then since triangle BFG is isosceles, BF is also 1 unit in length. Since triangle ABF is also isosceles, AF is also 1 unit in length. Let x denote the length of FG. Then since the triangles are similar we get the following ratio and proportion
which is the definition of the proportion of the golden mean.
To solve for x, clear denominators
Remove parentheses
Transpose
And use the quadratic formula
or
If we take the negative square root, we will get a negative answer, and since x is a distance in this case 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.
AFJ, ABG, and ACD are 36-72-72 triangles.
ABF and ABC are 36-36-108 triangles.