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To solve the given proof, we need to verify the ratio \(\frac{EG}{FG}\) using the relationship provided:

\[
\frac{EG}{FG} = \frac{AG \cdot CD + GP \cdot BD}{AG \cdot BD + GP \cdot CD}
\]

### Step-by-Step Analysis:

1. **Geometrical Setup:**
   - \(A, B, C, D, E, F, G, P\) represent various points in the triangle.
   - The lines drawn create smaller triangles inside, including \(AG, BD, CD, EG, FG, GP\), which will be involved in the proof.

2. **Key Ratios to Focus On:**
   - \(\frac{EG}{FG}\) must be derived from the relative positioning of points \(E\) and \(F\) on line \(AB\), and how point \(G\) relates to points \(P\) and \(A\).

3. **Ceva’s Theorem or Menelaus Theorem:**
   - This problem can likely be solved using **Menelaus' Theorem**, which deals with collinear points and transversals intersecting triangles.
   - We will apply Menelaus' theorem to triangle \(ABD\) with transversal \(EFG\) passing through points \(A, G, B\).

### Menelaus' Theorem Application:
Menelaus' theorem states that for a transversal intersecting the sides of a triangle, the following relationship holds:

\[
\frac{AE}{EB} \cdot \frac{BG}{GD} \cdot \frac{DP}{PA} = 1
\]

We can set up similar relationships for the triangle to compute \(EG/FG\).

4. **Combining Relations:**
   By substituting the relevant line segments and simplifying, you can verify if the given expression holds true.

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Would you like me to expand on the detailed application of Menelaus' theorem or explore another geometric theorem? 

### Related Questions:
1. How can Ceva's theorem help in solving geometric proofs involving ratios?
2. Can Menelaus' theorem be applied in all triangle transversal problems?
3. How do you approach problems involving multiple line intersections?
4. What is the significance of collinearity in triangle geometry?
5. How do ratios of line segments help in proving properties of triangles?

**Tip:** Menelaus' and Ceva's theorems are fundamental in triangle geometry, useful for solving complex ratios.

Prove the ratio \(\frac{EG}{FG} = \frac{AG \cdot CD + GP \cdot BD}{AG \cdot BD + GP \cdot CD}\)

Learn how to prove the ratio \(\frac{EG}{FG}\) using the relationship \(\frac{AG \cdot CD + GP \cdot BD}{AG \cdot BD + GP \cdot CD}\). This geometry problem explores line segments, ratios, and key theorems like Menelaus' and Ceva's theorems. Suitable for Grades 10-12.

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