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To solve this problem, let's break it down into steps based on the geometric properties given:

### Problem Summary:
- The figure shows a semicircle divided into four sections of equal area.
- \( A \) is a point on the diameter of the semicircle.
- We are asked to prove that \( A \) is the center of the circle and to find the angle \( \theta \).

### Step 1: Understanding the Diagram
- The semicircle has a diameter that is divided by a perpendicular from point \( A \), and another line drawn from \( A \) to the circumference forms the angle \( \theta \).
- The perpendicular from \( A \) meets the semicircle at one point, and a line from \( A \) to the opposite side of the semicircle also makes an angle \( \theta \) with the diameter.

### Step 2: Analyzing Equal Areas
- If the semicircle is divided into four sections of equal area, the line from \( A \) perpendicular to the diameter and the line forming \( \theta \) must have special geometric relationships.

### Step 3: Proving \( A \) is the Center of the Semicircle
1. **Area Considerations**: 
   - Since the semicircle is divided into four equal areas, each quarter-section must have an equal area.
   - The total area of the semicircle is \( \frac{1}{2} \pi r^2 \), so each section has an area of \( \frac{1}{8} \pi r^2 \).

2. **Symmetry Argument**: 
   - The symmetry of the semicircle and the equal area division suggests that \( A \) must be at the midpoint of the diameter for the areas to be equal, implying \( A \) is the center of the circle.

### Step 4: Finding \( \theta \)
- The triangle formed by the diameter, the perpendicular line, and the line forming \( \theta \) is a right triangle.
- Given that \( A \) is the center and the angle subtended by a diameter is \( 90^\circ \), the angle \( \theta \) must be \( 45^\circ \). This comes from the fact that in a semicircle, an inscribed angle subtending the diameter is always \( 90^\circ \).

### Conclusion:
- **Proof**: \( A \) is indeed the center of the semicircle due to the symmetry and equal area division.
- **\( \theta \)** is \( 45^\circ \).

Would you like more details on any part of the solution? Here are 5 relative questions for further exploration:

1. How can the area of segments in a semicircle be computed?
2. What are the properties of angles subtended by the diameter of a semicircle?
3. How does symmetry play a role in geometric proofs?
4. Can you derive \( \theta \) using trigonometric identities?
5. How do you prove that dividing the semicircle into equal areas implies \( A \) is the center?

**Tip**: When dealing with geometric proofs, symmetry and area calculations are often key to simplifying the problem and arriving at a solution.

Explore a geometric problem involving a semicircle divided into four equal areas and the calculation of a specific angle. This detailed solution uses concepts of semicircles, angle properties, and symmetry in geometric proofs. Suitable for Grades 11-12.

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