What is the best way to inscribe a golden rectangle into a pentagon? Do more golden ratios emerge?

Question

Below I drew a golden rectangle in a pentagon in Adobe Illustrator.

What would be the best way to inscribe a golden rectangle into a pentagon as shown in the figure below in a mathematical manner? If the length of the sides of the pentagon are 1, what is the length and width of the golden rectangle? How do the sides of the rectangle relate to the pentagram? What would be the best way to inscribe the golden rectangle in a perfect manner, such as in geogebra? I'm trying to figure it out! Thanks so much!

What is the ratio of A/B and C/D? (These do not seem to be golden ratios.)

Do more golden ratios emerge elsewhere? Is there anything golden about the triangles produced around the edges?

Thanks so much!!

Answer 1

With respect to incribing a golden rectangle into a pentagon, see the following figure:

First draw the pentagon $ABCDE$ and the golden rectangle $ECGF$.

After that, through homothety inscribe the Golden rectangle $KLIH$.