Stuck on a Geometry Problem 
$ABCD$ is a square, $E$ is a midpoint of side $BC$, points $F$ and $G$ are on the diagonal $AC$ so that $|AF|=3\ \text{cm}$, $|GC|=4\ \text{cm}$ and $\angle{FEG}=45 ^{\circ}$.  Determine the length of the segment $FG$.

How can I approach this problem, preferably without trigonometry?


Ok after some calculus, I figured out to solve this problem. Let's draw the vertical V that goes through E, and call $\alpha$ and $\frac{\pi}{4}-\alpha$ the angles we get on the left and right of the 45° angle cut by V and call c the length of the side of the square. We can then do some trigonometry to get :


*

*$c \cdot tan(\alpha) - (c-\frac{3}{\sqrt{2}}) = c \cdot tan(\alpha)$ (Continue the line EF to cut AD)

*$ tan(\frac{\pi}{4}-\alpha) = c \cdot \frac{\sqrt{2}}{8}-1$ (G is $\frac{4}{\sqrt{2}}$ away vertically from the right side and $\frac{c}{2}-\frac{4}{\sqrt{2}}$ away horizontally from the middle)


With a little trigo, we have 2 equations in $c$ and $tan(\alpha)$ that we can solve, which allow us to solve the whole problem. This makes me think that there is no "simple" answer without trigo :( 
Edit : Well, seems there was a simple answer afterall. Really nice proof, I'm glad I was wrong !
