Basic geometry problem question (Segment intersections?) The Problem

So, I was able to solve this problem, but what I'm curious of is once we solve the variables can I extend point F to line BC making Triangle ABX? I imagine an extension of point F would have to lie on BC at some point and intersect it. And because we're told that the x variables are part of a tri-section, each angle would have to be the same measure. 
When I first saw the problem I wanted to extend it and try and prove triangle ABX might be isosceles, but my instructor informed me that it was not possible, which I can see why now. But what about AFTER the variables are discovered, shouldn't that allow us to see where line AF would extend to? 
Or, would I need this shape to lie on a coordinate plane to be able to determine all this? Thanks in advance! 
Edit to clarify: I want to determine where the extension of segment AF would intersect segment BC making the Triangle ABX (where X is the new point extended from segment AF). Is it even possible to do so? 

If $AF\cap BC=\{X\}$ then $X$ can be placed not on segment $BC$.
Also, $x$ changes and you can not prove, what you want.
