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

Let $ABC$ be arbitrary triangle and $(\Omega)$ be an arbitrary circumconic of $ABC$ let $A'B'C'$ is its tangential triangle of $ABC$ respect to $(\Omega)$. Let $BB'$ meet $AC$ at $D$ and $CC'$ meet $AB$ at $E$, let $DE$ meet the circumconic at $F$. A line through $F$ and parallel to $B'C'$ meets $AB$, $AC$ at $H$, $G$ (see Figure) then: $$\frac{HG}{GF}=\frac{\sqrt{5}+1}{2}.$$