I am looking for a proof of an amazing discovery as follows:

Let $ABC$ be arbitrary triangle and $(\Omega)$ be arbitrary circumconic of $ABC$ let $A'B'C'$ is the its tangential triangle of $ABC$ respect to $(\Omega)$. Let $BB'$ meets $AC$ at $D$ and $CC'$ meets $AB$ at $E$, let $DE$ meets the circumconic at $F$. A line through $F$ and parallel to $B'C'$ meets $AB, AC$ at $H, G$ (see Figure) then:

$$\frac{HG}{GF}=\frac{\sqrt{5}+1}{2}$$