I really like your solution since it directly calculates the area.
My approach was similar but less direct. I called the area of the pink triangle x, and using the fact that the pink triangle and BEF are similar, with the base of BEF being 1/2 of the pink base, I got 1/2+1/4-x/4+x=1.
True confession: I initially, carelessly, used x/2 as the area of BEF. The result of x=1/2 didn't pass the small test, though. :-)