Thank you for the kind words.
Of course I should have mentioned this. The backward step if not obvious is standard. It's very much like that for Ceva's theorem:
Let there be three points such that AF/BF * BD/CD * CE/AE = 1 holds. Assume on the contrary that the points are not collinear. Pick up any two. Say D and E. Draw the line DE and find its intersection F' with AB. Then by the "forward" step AF'/BF' * BD/CD * CE/AE = 1. From which AF'/BF' = AF/BF. By subtracting 1
from both sides one gets AB/AF' = AB/AF, from which F' = F.
I have inserted this paragraph into the proof. Many thanks for pointing out this omission.
All the best,