4. In Project 3 we can see that there are many similar triangles in the pentagram.

In particular,

both Δ*AFI* and Δ*ABI* are 36-72-72 triangles, so they
are similar. If we let *AI* represent 1 unit of length, then since
Δ*AFI* is isosceles, *BF* is also 1 unit in length. Since
Δ*ABF* is also isosceles, *AF* is also 1 unit in length. Let *x* denote the length of *FG*. Then since

Δ*AFI* ∼ Δ*BAI*

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, 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.

Δ*AFJ*, Δ*ABG*, and Δ*ACD* are 36-72-72 triangles.

Δ*ABF* and Δ*ABC* are 36-36-108 triangles.

Both types are called **Golden Triangles**. There are thirty golden triangles in a pentagram.