Find ratio of sides in touching pentagon problem

Question

I need to find the ratio d/s, I suspect that is the golden ratio because of this: https://www.cut-the-knot.org/pythagoras/cos36.shtml, but I'm not supposed to use trigonometry

I showed that the length of the longer side in the isoceles triangle in the right triangle is s(2cos(36)) then using similar triangles showed that d/s is (2cos36+1)/(2cos36), but I want to do this without using trigonometry

Answer 1

Let $DE$ be a bisector of $\Delta BCD$.

Thus, $$s=BD=DE=CE,$$ $$CD=CB=d,$$ $$BE=d-s$$ and since $\Delta DBC\sim\Delta BED,$ we obtain $$\frac{d}{s}=\frac{s}{d-s},$$ which gives $$\frac{d}{s}=\frac{1+\sqrt5}{2}.$$

Answer 2

Let $F$ be the vertex on the first pentagon connected with $A$ by $d$. Use angle chasing to prove that $AF=AD=FD$. Finally, observe the result of Ptolomey’s Theorem on the quadrilateral $ABDF$ $$ds+s^2=d^2$$ to show that $$\frac{d}{s}=\varphi$$