using parallel line theorems and side angle side ect. in triangle acd a line is drawn from point b on ac and point i on ad. another triangle fliped vertically and joined at d is named fdh, a line is drawn from point e on fd and point g on fh. ab = fg, ai = fe, ac || fh prove that bi || eg /-\ \-/

I think the problem needs to be more carefully worded. I want to show you three diagrams that seem to satisfy your description, but only in one of them is bi || eg.

First diagram

Harley