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The question involves a parallelogram \(ABCD\) with extended lines and points forming triangles and intersections. The problem specifically asks to prove certain proportionalities in triangles formed by these extensions.

**Let's address each part of the question:**

### 4.1 Prove that \(\frac{EP}{ER} = \frac{DQ}{DP}\)

Given:
- \(ABCD\) is a parallelogram.
- \(P\) is a point on \(BD\) produced, and \(PC\) produced intersects \(AB\) produced at \(Q\).
- \(AB\) produced meets \(PC\) at \(Q\).

**To Prove:** \(\frac{EP}{ER} = \frac{DQ}{DP}\)

Since \(ABCD\) is a parallelogram, \(AB \parallel CD\) and \(AD \parallel BC\).

### 4.2 Prove that \(\frac{PN}{NR} = \frac{PN}{NQ}\)

**Given:**
- \(P\), \(N\), \(R\), and \(Q\) are points on the line segment.

**To Prove:** \(\frac{PN}{NR} = \frac{PN}{NQ}\)

This involves a similar setup, relying on parallel lines and transversals.

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To proceed, we'll use properties of similar triangles, parallel lines, and the basic proportionality theorem (also known as Thales's theorem). However, due to the image quality, it's challenging to read the specifics. If possible, could you provide a clearer image or type out the text?

**Do you want a detailed explanation for each part, or is there a specific part you need more help with?**

### Follow-up Questions

1. What are the definitions of similar triangles, and how are they used in proofs?
2. What is the significance of parallel lines in proving proportionalities?
3. How do we use the properties of a parallelogram in geometric proofs?
4. Can you provide a step-by-step approach to proving triangles are similar?
5. How do we apply Thales's theorem in this context?
6. What is the role of transversal lines in creating proportional relationships?
7. How can we identify corresponding sides in similar triangles?
8. What are the common errors to avoid when proving proportionalities in geometry?

**Tip:** Always label diagrams clearly and keep track of all given information, especially when working with extensions of geometric figures.

Learn how to prove proportional relationships involving triangles formed by extended lines in a parallelogram. This problem explores geometric concepts like proportional relationships, parallel lines, and the properties of parallelograms. Suitable for high school students studying geometry.

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