We are given a triangle $ABC$ with the following ratios for segments intersecting the triangle:

• $AE:CE = 1:2$
• $AF:BF = 5:4$
• $BD:CD = 3:2$

We are tasked with finding the ratio of $AG:DG$, where $AD$ is a cevian and $EF$ is a transversal intersecting $AD$ at $G$.

Step 1: Use Menelaus' Theorem

To solve this, we can apply Menelaus' theorem, which relates the ratios of segments in a triangle when a transversal intersects the sides of the triangle.

Menelaus' theorem states that for a transversal intersecting the sides of a triangle (or their extensions), the following equation holds:

$\frac{AE}{CE} \cdot \frac{CF}{BF} \cdot \frac{BD}{AD} = 1$

Here, the transversal $EF$ intersects triangle $ABC$, and we can label the segments involved in the ratio:

• $\frac{AE}{CE} = \frac{1}{2}$
• $\frac{AF}{BF} = \frac{5}{4}$
• $\frac{BD}{CD} = \frac{3}{2}$

Step 2: Applying the Ratios

Using Menelaus' theorem for the transversal $EF$ and cevian $AD$, we have:

$\frac{AE}{CE} \cdot \frac{AF}{BF} \cdot \frac{BD}{CD} = 1$

Substitute the given ratios into the equation:

$\frac{1}{2} \cdot \frac{5}{4} \cdot \frac{3}{2} = 1$

Calculate the product:

$\frac{1 \times 5 \times 3}{2 \times 4 \times 2} = \frac{15}{16}$

This does not satisfy Menelaus' theorem because the result is not equal to 1. Therefore, the cevian ratio $\frac{AG}{DG}$ must correct the imbalance.

Step 3: Adjusting for the Ratio of $\frac{AG}{DG}$

To correct this, we introduce the ratio $\frac{AG}{DG} = x$. Thus, the equation becomes:

$\frac{1}{2} \cdot \frac{5}{4} \cdot \frac{3}{2} \cdot x = 1$

Solve for $x$:

$\frac{15}{16} \cdot x = 1$

$x = \frac{16}{15}$

Thus, the ratio of $AG$ to $DG$ is:

$\boxed{\frac{AG}{DG} = \frac{16}{15}}$

Conclusion

The ratio of $AG$ to $DG$ is $\frac{16}{15}$.

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Would you like further details or explanations?

Here are 5 related questions to consider:

1. How can Menelaus' theorem be used for other transversals in triangles?
2. What is the significance of cevians in triangle geometry?
3. How do ratios help in determining the position of points within geometric shapes?
4. Can Menelaus' theorem be applied to quadrilaterals?
5. How would the solution change if the transversal did not intersect the cevian?

Tip: When using Menelaus' theorem, always ensure that the ratios from the transversal points are calculated correctly to maintain the balance in the equation.