This construction relies heavily on the homework problem on special isosceles triangles whose sides are in the proportion of the golden ratio and whose angles are 36-72-72 degrees, which we now call golden isosceles triangles.
If a star pentagon (pentagram) is inscribed in a circle, we have already seen that the angle of the star point is 36 degrees, so the figure is full of golden isosceles triangles.
Given a side a, a length b can be constructed so that b/a is the golden ratio = (1 + sqrt(5))/2 (from homework). Then the vertices of a triangle with sides a, b, b can be constructed using the compass.
Such a triangle can be constructed using SSS.
The golden triangle in 2 gives 3 of the 5 vertices. The other 2 can be constructed by constructing triangles or parallelograms from the first 3 points.
Additional Construction: Pentagon from the center
There is a construction for the regular pentagon given in B&B, but the construction there is not an answer to this question, since that construction takes as given the center O of the pentagon and one vertex A. It is simpler to use the class handout from Monday 10/23 as a guide.
Construct a Golden Rectangle
Let ABCD be a rectangle, with AB < BC. Then there is a point E on segment AD and a point F on segment BC such that CDEF is a square and BFEA is a rectangle. We say that ABCD is a golden rectangle if BFEA is similar to ABCD. If ABCD is a golden rectangle, there is only one possible ratio BC/AB. Compute this number (exactly). Give your reasoning.
Construct an example of such a rectangle.