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


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


x2 + x - 1 = 0

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.