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](https://mathworld.wolfram.com/TangentialTriangle.html) 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:*
 $$\frac{HG}{GF}=\frac{\sqrt{5}+1}{2}.$$

[![Geometric arrangement described in the text][1]][1]


  [1]: https://i.sstatic.net/j7qjj.png