6. (10 points) As shown in the figure, the area of triangle $ABC$ is $1, D O: O B=1: 3, E O: O A=4: 5$, then the area of triangle $D O E$ is $\qquad$ .
【Analysis】It is not difficult to find that when the height is the same, the ratio of the areas is equal to the ratio of the base sides, which is the application of the proportional relationship between lines. The fraction of the area of $\triangle D O E$ to the total area of the triangle can be found using the method of fractions.

【Solution】Solution: According to the problem, as shown in the figure:
Let the area of triangle $D O E$ be $4 x$. By the proportional relationship, it is not difficult to derive that the areas of the other three parts in the figure are $5 x, 12 x, 15 x$, respectively. Let the area of triangle $D C E$ be $y$,
In $\triangle D B C$ and $\triangle A B C$, we have $\frac{\mathrm{CE}}{\mathrm{BE}}=\frac{\mathrm{y}}{4 \mathrm{x}+12 \mathrm{x}}=\frac{\mathrm{y}+4 \mathrm{x}+5 \mathrm{x}}{12 \mathrm{x}+15 \mathrm{x}}$.
Thus, $y=\frac{144}{11} x$.
Therefore, the area of $\triangle D O E$ is $\frac{4}{4+5+12+15+\frac{144}{11}}=\frac{11}{135}$.
Since the area of $\triangle A B C$ is 1.
Therefore, the area of $\triangle D O E$ is $\frac{11}{135}$.
Hence, the answer is: $\frac{11}{135}$.