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

x(x + 1) = 1

Remove parentheses

x2 + x = 1

Transpose

x2 + x - 1 = 0

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

FJ/AF

AF/AB

AB/AD

AB/AC

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.

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