In regular octagon $ABCDEFGH$, $M$ and $N$ are midpoints of $\overline{BC}$ and $\overline{FG}$ respectively. Compute $[ABMO]/[EDCMO]$. ($[A

O will be the center of the octagon

Draw OB, OC  and OD

Note that   area OAB  =  area  ODC =  area ODE

And area OBM  =  1/2 area OAB

And area OCM  = area OBM  =  1/2 area OAB

So

area  EDCMO  =   area ODE + area ODC  + area OCM

Substituting, we have that

area EDCMO  =  area OAB + area OAB + 1/2area OAB   =  2.5 areaOAB

And area ABMO  =  area OAB + area OBM

Substituting, we have that

area ABMO  =  area OAB + 1/2 area OAB   =   1.5 area OAB

So

[ ABMO] / [ EDCMO]  =

[ 1.5 area OAB ] / [ 2.5 area OAB ]  =

[1.5] / [ 2.5]  =

3 / 5