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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

$$Area_{AEG} : Area_{AEFO}$$

[parallelogram


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.

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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 $$ {Area_{AEG}\over Area_{AEFO}}= {(1/2)Area_{EOA}\over(4/3)Area_{EOA}}={3\over8}. $$

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  • $\begingroup$ Your answer is helping me to resolve it, however, the ratio between similar triangles is not the same than the ratio between their areas. $\endgroup$ – Pichi Wuana May 10 '16 at 19:36
  • $\begingroup$ You know that there need to be equal angles between similar triangles... $\endgroup$ – Pichi Wuana May 10 '16 at 19:45
  • $\begingroup$ I'm not using similar triangles. If triangles $EOF$ and $EOA$ share the same basis and their altitudes are in the ratio $1:3$, then their areas are in the same ratio. $\endgroup$ – Aretino May 10 '16 at 19:51
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    $\begingroup$ OK, I've expanded my answer. $\endgroup$ – Aretino May 10 '16 at 21:27
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    $\begingroup$ You are right: I inadvertedly wrote the inverse fraction. Corrected now. $\endgroup$ – Aretino May 10 '16 at 22:19

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