Golden Ratios in Regular Polygons

Here’s the first 3: for the triangle, square and pentagon.

Equilateral Triangle 

The triangle ABC is similar DEC so $ \frac{AC}{BC} = \frac{a}{b} = \frac{CD}{CE} = \frac{a+b}{a} = \phi $

Square

In square ABCD M,N,P,Q are midpoints of the sides, as shown below; F the center of the square. Then circle (AP) on AP as a diameter, cuts NF (point X) in Golden Ratio. Chord AX cuts MF in Golden Ratio:

Pentagon