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We are given a triangle \( \Delta PQR \), where \( PQ = PR \), and \( M \) is a point on \( PQ \). The line through \( M \) intersects \( PR \) at \( O \) and \( QR \) extended at \( N \). We are asked to prove that \( PO > PM \).

### Proof:

1. **Given:**
   - \( \Delta PQR \) is an isosceles triangle with \( PQ = PR \).
   - \( M \) is a point on \( PQ \), and the line through \( M \) intersects \( PR \) at \( O \) and \( QR \) extended at \( N \).
   - We need to prove that \( PO > PM \).

2. **Step-by-step reasoning:**

   - Since \( PQ = PR \), triangle \( PQR \) is isosceles.
   - The points \( M \) and \( O \) are both on the line segment \( PQ \) and \( PR \), respectively.
   - Notice that \( O \) lies on \( PR \), and the line through \( M \) intersects \( QR \) at \( N \), extending beyond \( R \).
   - Now, let's focus on triangles \( POM \) and \( PMN \):
     - Since the point \( M \) is on \( PQ \), and the line through \( M \) intersects both \( PR \) and \( QR \), the segments \( PM \) and \( PO \) are part of a geometric setup where the point \( O \) is closer to \( P \) than \( M \), as the line from \( M \) extends and cuts at \( N \).

3. **Using Triangle Inequality:**
   - In \( \Delta POM \), applying the triangle inequality theorem, we get:
     \[
     PO + OM > PM
     \]
     Since \( OM \) is a positive distance, we can conclude:
     \[
     PO > PM
     \]
     This proves that the segment \( PO \) is longer than \( PM \).

### Conclusion:
Thus, we have proven that \( PO > PM \) using the triangle inequality theorem applied to \( \Delta POM \).

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Would you like more details or further explanation on any part of the proof?

Here are five related questions that might interest you:
1. How does the isosceles property of \( \Delta PQR \) affect the line segment relationships in the triangle?
2. Can we use any other theorem to prove that \( PO > PM \)?
3. How does the extension of line \( QR \) at \( N \) influence the triangle's geometry?
4. What role does the point \( O \) play in the inequality \( PO > PM \)?
5. Can this proof be applied to other configurations of isosceles triangles?

**Tip:** Always consider applying the triangle inequality theorem when comparing the lengths of sides in geometric figures.

In ΔPQR, PQ = PR. M is a point on PQ and the line through M intersects PR at O and QR extended at N. Prove that PO > PM.

Learn how to prove that PO > PM in an isosceles triangle ΔPQR, where PQ = PR, using the Triangle Inequality Theorem. This detailed solution explains key geometry concepts suitable for students in Grades 10-12.

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