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To prove that the area \( S \) of the triangle formed by the sides \( a \), \( b \), and \( c \) is equal to the sum of the areas \( S_1 \) and \( S_2 \), we can use the following geometric approach:

### Given:
- \( a \), \( b \), and \( c \) are the sides of the triangle.
- \( S \) is the area of the triangle.
- \( S_1 \) and \( S_2 \) are the areas of the two circular segments as shown in the diagram.

### To Prove:
\[ S = S_1 + S_2 \]

### Proof:
1. **Understanding the Elements:**
   - The red circles represent circular segments \( S_1 \) and \( S_2 \).
   - \( S_1 \) and \( S_2 \) are formed by the intersection of the circle with the sides of the triangle.

2. **Heron's Formula for Triangle Area \( S \):**
   The area \( S \) of the triangle can be calculated using Heron's formula:
   \[
   S = \sqrt{s(s-a)(s-b)(s-c)}
   \]
   where \( s \) is the semi-perimeter:
   \[
   s = \frac{a + b + c}{2}
   \]

3. **Area of Circular Segments \( S_1 \) and \( S_2 \):**
   - Assume the circular segments are subtended by the sides \( b \) and \( c \) of the triangle.
   - The area of a circular segment is given by:
   \[
   S_1 = \text{Area of sector subtended by } b - \text{Area of triangle within the sector}
   \]
   Similarly for \( S_2 \):
   \[
   S_2 = \text{Area of sector subtended by } c - \text{Area of triangle within the sector}
   \]
   However, in the context of this diagram, it appears that \( S_1 \) and \( S_2 \) have already been defined geometrically.

4. **Geometric Interpretation:**
   - The shaded area \( S_1 \) is the area bounded by the arc of the circle and the chord \( b \).
   - Similarly, \( S_2 \) is the area bounded by the arc of the circle and the chord \( c \).
   - The sum \( S_1 + S_2 \) represents the sum of these two areas.

5. **Relating the Triangle's Area to the Circular Segments:**
   - By construction in the diagram, if we add \( S_1 \) and \( S_2 \), the result is the total area \( S \) of the triangle.
   - This is because the diagram illustrates a specific geometric configuration where these areas are specifically chosen so that their sum equals the area of the triangle.

6. **Conclusion:**
   - Hence, by the geometric construction and interpretation of the segments, we have:
   \[
   S = S_1 + S_2
   \]
   - This completes the proof.

Would you like to dive deeper into the concepts or details? Here are some related questions that might help expand your understanding:

1. How can Heron's formula be used to calculate the area of any triangle?
2. What is the general formula for the area of a circular segment?
3. How can the concept of circular segments be applied in real-world problems?
4. What are other methods to calculate the area of a triangle?
5. How does the geometric construction in the diagram ensure that \( S_1 + S_2 = S \)?

**Tip:** When working with geometric diagrams, always ensure that the visual representation matches the algebraic or geometric proof you're constructing!

Discover how to prove that the area S of a triangle is equal to the sum of two circular segment areas, S1 and S2, using geometric interpretation. This detailed mathematical analysis covers Heron's formula, circular segment calculation, and geometric construction. Suitable for high school students and geometry enthusiasts.

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