6. Construction of the pentagon.

Start with a circle. Let the radius of the circle denote 1 unit.

Let O be the center of the circle, and let BC be a diameter perpendicular to OA. Let M be the midpoint of OA, and draw the circle centered at M which goes through O and A. Next, connect M and B.

Let D be the point where MB intersects the circle centered at M going through O and A. Let us look at triangle OMB.

We can use the Pythagorean Theorem to find c.

If we let BD = x,

then

which is the proportion of the golden mean. Draw the circle centered at B of radius x

Let the points where this circle intersects the original circle be
E and F. Then in the isosceles triangle OBE, the ratio of the base to
the equal sides is x, so we know that angle BOE is a 36^{o}
angle. It follows that angle FOE is a 72^{o} angle which is
1/5 of the circle. If we draw circles at F and E whose radius is EF,

Let the points where these circles intersect the original circle be G and H. Then EFGCH is a regular pentagon.