<s>[INST] 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?

 [/INST] Drop perpendiculars $FH, EI$ and $FJ$ and label $BH=y$, $EH=z$ and $\angle GEI=\alpha$:
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Use the similarity of $\triangle FEH$ and $\triangle EIG$:
$$\frac{GI}{HE}=\frac{EI}{FH} \Rightarrow \\
\frac{4-CI}{z}=\frac{CI}{y+2z}\Rightarrow \\
\frac{4-\frac{y+z}{\sqrt{2}}}{z}=
\frac{\frac{y+z}{\sqrt{2}}}{y+2z}\Rightarrow \\
\frac{4\sqrt{2}-y-z}{z}=\frac{y+z}{y+2z}\Rightarrow\\
3z^2+(4y-8\sqrt{2})z+y^2-4\sqrt{2}y=0 \stackrel{y=\frac3{\sqrt{2}}}{\Rightarrow} \\
6z^2-4\sqrt{2}z-15=0 \Rightarrow \\
z=\frac3{\sqrt{2}}.$$
Hence:
$$x+7=2(y+z)\sqrt{2}=12 \Rightarrow x=5.$$
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