Curious about the theorem

Thank you very much for the help emakarov! Please forgive me if I'm being dense, but the claim that angle FCA = angle ABC is difficult for me to accept given that F is an *arbitrary *point on m, only constrained such that it lies to the left of C. Because angle ABC is fixed once constructed, my wee brain cannot find a satisfactory way to believe the proposition.

I searched a bit for the theorem you mentioned, but didn't get far because I really don't know what I might search for, i.e. what to guess that the theorem might be called.

Notwithstanding all that, thanks for your help. (Bow)