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I see the mathematical problems in the image you uploaded. Let’s focus on the 13th problem first:

**Problem 13**:  
_"D is a point on the side BC of a triangle ABC such that ∠ADC = ∠BAC. Show that \(CA^2 = CB \cdot CD\)."_

This is a classic problem that involves properties of triangles, particularly the *angle bisector theorem* and the use of *similar triangles*. Here is a step-by-step approach to prove this statement:

### Proof:
1. **Identify similar triangles**:  
   Since \( \angle ADC = \angle BAC \), triangles \( \Delta ADC \) and \( \Delta ABC \) are similar by the **AA (Angle-Angle) similarity criterion**, because they also share the angle \( \angle A \).

2. **Apply the similarity ratio**:  
   Since \( \Delta ADC \sim \Delta ABC \), corresponding sides are proportional. Hence, we can write:
   \[
   \frac{CA}{CB} = \frac{CD}{CA}
   \]

3. **Cross-multiply the proportion**:
   By cross-multiplying the terms, we get:
   \[
   CA^2 = CB \cdot CD
   \]
   
Thus, the proof is complete. This relation shows how the square of one side (in this case, \(CA\)) relates to the product of two other segments on the opposite side.

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Would you like further clarification or additional details on any step?  
Here are five related questions to expand your understanding:

1. What is the AA (Angle-Angle) criterion for similar triangles?
2. How does the Angle Bisector Theorem relate to triangle similarity?
3. Can this result be generalized to other types of triangles?
4. What are other methods to prove triangle similarity?
5. How can this result be applied to find specific lengths in a triangle?

**Tip**: Understanding triangle similarity helps with solving problems in both geometry and trigonometry, especially when working with proportions and relationships between sides.

D is a point on the side BC of a triangle ABC such that ∠ADC = ∠BAC. Show that CA^2 = CB · CD.

Explore how to prove that CA^2 = CB · CD in triangle ABC using the Angle-Angle (AA) similarity criterion. This problem demonstrates key concepts of triangle similarity and proportionality of sides, making it suitable for students in Grades 9-11.

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