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.