I have a parallelogram $ABCD$. $E$ is center of $AD$. $O$ is center of $AC$ and center of $DB$. $F$ is the intersection point between $CE$ and $DO$. Point $G$ is the intersection between $EO$ and $AF$. $H$ is the middle of $DC$.
I put variables to different lines in the figure to show the ratios I found.
I need to find the ratio
$$Area_{AEG} : Area_{AEFO}$$
I don't have an idea on how to find it. I know how to find ratios like for example $$Area_{AEG} : Area_{ADH}$$ because they are similar triangles.
If $H$ is the intersection between $AF$ and $CD$, then $GF=(1/2)FH$ so that $GF=(1/3)GH$, that is: $GF=(1/3)AG$. If $AM$ and $FN$ are the altitudes of triangles $AEO$ and $FEO$ with respect to $EO$ as common base, then $AGM$ and $FGN$ are similar triangle and $FN=(1/3)AM$. It follows that $Area_{EOF}={1\over3}Area_{EOA}$.
On the other hand triangles $AEG$ and $AOG$ have equal bases $EG$ and $OG$, and share the same altitude $AM$. It follows that $Area_{AEG}=Area_{AOG}$, that is: $Area_{AEG}={1\over2}Area_{EOA}$.
In the end we have: $$ Area_{AEFO}=Area_{EOF}+Area_{EOA}={1\over3}Area_{EOA}+Area_{EOA}={4\over3}Area_{EOA} $$ and $$ Area_{AEG}={1\over2}Area_{EOA}, $$ so that $$ {Area_{AEG}\over Area_{AEFO}}= {(1/2)Area_{EOA}\over(4/3)Area_{EOA}}={3\over8}. $$