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

In particular,

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

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, 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.

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.

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