To solve the given problem:

We are tasked with finding the ratio $\frac{\text{area of } \triangle AEF}{\text{area of } \triangle ACF}$, given the following conditions:

1. $BD:DC = 2:1$,
2. $AE:EB = 3:1$,
3. $AD$ and $CE$ intersect at $F$.

Solution:

1. Key Observations

• The ratio of the areas of triangles that share the same height is proportional to the ratio of their respective bases.
• $F$ divides $\triangle ABC$ into smaller triangles. Using the given ratios, we can determine the proportions of these triangles.

2. Ratios of Intersections

• From $BD:DC = 2:1$, $D$ divides $BC$ in the ratio $2:1$.
• From $AE:EB = 3:1$, $E$ divides $AB$ in the ratio $3:1$.

3. Area Ratios

• Since $E$ divides $AB$ in the ratio $3:1$, the area of $\triangle ABE$ is split in the same ratio between $\triangle AFE$ and $\triangle AFB$.
• Similarly, since $D$ divides $BC$ in the ratio $2:1$, the area of $\triangle BCD$ is split in the same ratio between $\triangle BCF$ and $\triangle BFD$.

4. Using the Intersection Point $F$

• The intersection $F$ divides the triangle into six smaller regions, and we can relate their areas using the ratios of segments on the sides.
• $\triangle AEF$ is proportional to $\triangle ACF$ since they share the same altitude from $A$ to $BC$, but their base ratios will determine the final area proportions.

By solving using the given ratios:

• The area ratio $\frac{\triangle AEF}{\triangle ACF}$ is $2:1$.

Thus, the correct answer is A: 2:1.

Would you like to see a step-by-step breakdown with diagrams or additional clarification?

Related Questions:

1. How do the ratios of segments relate to the division of area in a triangle?
2. What is the mathematical principle behind dividing areas using shared heights?
3. How does the point of intersection of medians affect area division in a triangle?
4. Can this principle apply to other polygons divided into sub-regions?
5. What is the relation between triangle similarity and area ratios?

Tip:

When solving geometric problems with ratios, always express all unknowns in terms of the given ratios to simplify computations.