Given $A(r,0),B(r\cos 72^{\circ},r\sin 72^{\circ}),C(-r\cos 36^{\circ},r\sin 36^{\circ}),D(-r\cos 36^{\circ},-r\sin 36^{\circ}),E(r\cos 72^{\circ},-r\sin 72^{\circ})$. Take $F(r\cos \alpha,r\sin \alpha)$. $FD=\sqrt{(r\cos \alpha+r\cos 36^{\circ})^{2}+(r\sin \alpha+r\sin 36^{\circ})^{2}}$. $FD=\sqrt{2+2\cos(\alpha-36^{\circ})}=2\cos \frac{\alpha-36^{\circ}}{2}$. Idem $FE=2\sin \frac{\alpha+72^{\circ}}{2}$ and $FC=2\cos \frac{\alpha+36^{\circ}}{2}$. We calculate $\frac{FD}{FE+FC}=\frac{\cos \frac{\alpha-36^{\circ}}{2}}{\sin \frac{\alpha+72^{\circ}}{2}+\cos \frac{\alpha+36^{\circ}}{2}}$. $\frac{FD}{FE+FC}=\frac{\cos \frac{\alpha-36^{\circ}}{2}}{\sin \frac{\alpha+72^{\circ}}{2}+\sin \frac{144^{\circ}-\alpha}{2}}$. $\frac{FD}{FE+FC}=\frac{\cos \frac{\alpha-36^{\circ}}{2}}{2\sin 54^{\circ}\cos \frac{\alpha-36^{\circ}}{2}}$. $\frac{FD}{FE+FC}=\frac{1}{2\sin 54^{\circ}}=\frac{1}{\frac{\sqrt{5}+1}{2}}=\frac{\sqrt{5}-1}{2}$.