It is nice that the golden ratio appears automatically when we are not looking for it. This is what happened to me when I was using GeoGebra and trying to solve a different problem that occurred to me:
Data: $C$ parabola Featured F and L, Point A moving freely on L, Point B represents Muscat-based F on L, Point M represents
Intersection of a F widget with C.
Required: Find the appropriate position for a point A which makes AM=FB.
Was it previously discovered or not? In any case, how do we prove that?