If `r,r_(1) ,r_(2), r_(3)` have their usual meanings , the value of `1/(r_(1))+1/(r_(2))+1/(r_(3))`, is

Options

A. 1
B. 0
C. `1/r`
D. none of these

Answer

To solve the problem step by step, we will analyze the given expressions and derive the required value systematically.

### Step-by-Step Solution:

1. **Understanding the Variables**:
We have \( r, r_1, r_2, r_3 \) where:
- \( r_1 = \frac{\Delta}{s - a} \)
- \( r_2 = \frac{\Delta}{s - b} \)
- \( r_3 = \frac{\Delta}{s - c} \)
Here, \( \Delta \) is the area of the triangle, and \( s \) is the semi-perimeter given by \( s = \frac{a + b + c}{2} \).

**Hint**: Identify the meanings of \( r, r_1, r_2, r_3 \) in the context of triangle geometry.

2. **Setting Up the Expression**:
We need to find the value of:
\[ \frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3} \]

3. **Substituting the Values**:
Substitute the values of \( r_1, r_2, r_3 \):
\[ \frac{1}{r_1} = \frac{s - a}{\Delta}, \quad \frac{1}{r_2} = \frac{s - b}{\Delta}, \quad \frac{1}{r_3} = \frac{s - c}{\Delta} \]
Therefore, we can write:
\[ \frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3} = \frac{s - a}{\Delta} + \frac{s - b}{\Delta} + \frac{s - c}{\Delta} \]

**Hint**: Substitute the expressions for \( r_1, r_2, r_3 \) into the equation.

4. **Combining the Fractions**:
Since all terms have a common denominator \( \Delta \), we can combine them:
\[ \frac{(s - a) + (s - b) + (s - c)}{\Delta} \]

5. **Simplifying the Numerator**:
Simplifying the numerator:
\[ (s - a) + (s - b) + (s - c) = 3s - (a + b + c) \]
Since \( a + b + c = 2s \), we have:
\[ 3s - (2s) = s \]

**Hint**: Use the relationship \( a + b + c = 2s \) to simplify the numerator.

6. **Final Expression**:
Now substituting back, we get:
\[ \frac{s}{\Delta} \]

7. **Relating to \( r \)**:
We know that \( r = \frac{\Delta}{s} \), hence:
\[ \frac{s}{\Delta} = \frac{1}{r} \]

8. **Conclusion**:
Therefore, we conclude that:
\[ \frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3} = \frac{1}{r} \]

Thus, the final answer is:
\[ \frac{1}{r} \]

### Final Answer:
\[ \frac{1}{r} \]