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Solve this qn based on the symmetric properties of circles. 

To solve this problem based on the symmetric properties of circles, let's analyze the diagram step by step:

1. **Given Information**:
   - The diagram shows a circle with several points marked on it (A, B, C, D).
   - The points on the circle are labeled as \( A \), \( B \), \( C \), and \( D \), with the line segments creating certain angles and triangle-like shapes.
   - The values of angles seem to be given or deduced from the circle's symmetry, with \( \angle BAC = 90^\circ \), and possibly an inscribed angle theorem in use.

2. **Symmetry and Circle Properties**:
   - **Inscribed Angles**: An important property of a circle is that the angle subtended by an arc at any point on the circumference is half the angle subtended by the same arc at the center. If we have an inscribed angle like \( \angle BAC \), it will be half of the central angle subtended by the same arc.
   
   - **Cyclic Quadrilateral**: If the points \( A, B, C, D \) form a quadrilateral inside the circle, then the opposite angles of this cyclic quadrilateral will sum up to \( 180^\circ \) (supplementary angles property).

   - **Right Angles**: If the angle \( \angle BAC = 90^\circ \), then according to the inscribed angle theorem, the arc \( BC \) must subtend a right angle.

3. **Potential Calculations**:
   If the problem involves calculating the angle \( \angle BDC \) (as suggested by a cyclic quadrilateral), we might use the fact that the sum of opposite angles in a cyclic quadrilateral equals \( 180^\circ \).

4. **Final Solution**: Based on symmetry and the inscribed angle theorem, as well as supplementary angles in cyclic quadrilaterals, the question might be asking for the angle values or other properties derived from these rules.

Would you like a detailed step-by-step breakdown of how to apply these properties to this specific circle and its angles?

Here are some follow-up questions related to the problem:

1. What is the exact value of \( \angle BDC \) in terms of the cyclic quadrilateral properties?
2. How can we use the inscribed angle theorem to calculate angles other than \( \angle BAC \)?
3. What other geometric properties can be derived from the symmetry of the circle?
4. If we know the length of one of the sides of a triangle formed by points on the circle, how can we calculate the other sides using the Law of Cosines or Law of Sines?
5. How does the property of perpendicularity (such as \( \angle BAC = 90^\circ \)) affect the other angles and lengths?

**Tip**: Always remember that the sum of opposite angles in a cyclic quadrilateral is \( 180^\circ \). This property is often crucial in solving problems involving inscribed circles and quadrilaterals.

Solve this question based on the symmetric properties of circles.

Explore the symmetric properties of circles with a focus on inscribed angles and cyclic quadrilateral theorems. This problem involves key geometric concepts and is suitable for students in Grades 9-11.

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