In parallelogram ABCD there is line through A intersects segment BD in E, segment CD in F, line BC in G, find ED if $FG:FE = 7$ and BE = 8
I tried to write a lot similarities, but I can't get $AE:EF$ anyway. I draw lines through E parallel to $AB$ and $AD$. I stuck, I don't know what to do next.
Say $AB = a$, $DF = b$, $AE =z$, $DE=y=?$, $EF =x$ and $FG=7x$.
Then since $ABE\sim FDE$ we have $${z\over x} = {8\over y} = {a\over b}$$ and since $ABG\sim FCG$ we have $${7x\over 8x+z} ={a-b\over a}$$ so $${7\over 8+{z\over x}} = 1-{b\over a}$$ so $${7 \over 8+{8\over y}} = 1-{y\over 8}$$ ... $y= 2\sqrt{2}$.