let $\widehat{AOE}=\widehat{MOC}=\alpha$ , $\widehat{EOD}=\widehat{BOM}=\beta$ , $\widehat{AOB}=\widehat{DOC}=y$ we have by sinus law $\frac{AE}{ED}=\frac{sin(\alpha)}{sin(\beta)}.(\frac{AO}{DO})$ by sinus law again in the triangle $OBM$ and $OMC$ we have : $\frac{sin(\alpha)}{sin(\beta)}=\frac{OB}{OC}$ since $M$ is the midelpoint of $BC$ , and so $\frac{AE}{ED}=\frac{AO.BO}{DO.OC}=\frac{S_{ABO}}{S_{DCO}}$ .