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,
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 36o angle. It follows that angle FOE is a 72o 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.