If the given figure, E and F are the centers of two identical circles. What is the ratio of area of triangle AOB to the area of triangle DOC?

$$\eqalign{ & \Delta DEM \sim \Delta DBN \cr & \therefore \frac{{EM}}{{BN}} = \frac{{DM}}{{DN}} \cr & \frac{r}{{2r}} = \frac{{DM}}{{DN}} \cr & \Rightarrow DM:MN = 1:1 \cr & {\text{Similarly }}MN:NC = 1:1 \cr & \frac{{{\text{ar}}{\text{. }}\Delta AOB}}{{{\text{ar}}{\text{. }}\Delta DOC}} = {\left( {\frac{1}{3}} \right)^2} = \frac{1}{9} = 1:9 \cr} $$