The exradii of a triangle `r_(1),r_(2),r_(3)` are in HP , then the sides a,b,c are

Options

A. in HP
B. in AP
C. in GP
D. none of these

Answer

To solve the problem, we need to show that if the exradii \( r_1, r_2, r_3 \) of a triangle are in Harmonic Progression (HP), then the sides \( a, b, c \) of the triangle are in Arithmetic Progression (AP).

### Step-by-Step Solution:

1. **Understanding Exradii**:
The exradii \( r_1, r_2, r_3 \) of a triangle can be expressed as:
\[ r_1 = \frac{\Delta}{s - a}, \quad r_2 = \frac{\Delta}{s - b}, \quad r_3 = \frac{\Delta}{s - c} \]
where \( \Delta \) is the area of the triangle and \( s \) is the semi-perimeter given by \( s = \frac{a + b + c}{2} \).

2. **Condition for Harmonic Progression**:
Since \( r_1, r_2, r_3 \) are in HP, we can use the property of HP which states that:
\[ \frac{1}{r_1}, \frac{1}{r_2}, \frac{1}{r_3} \text{ are in AP} \]
This means:
\[ 2 \cdot \frac{1}{r_2} = \frac{1}{r_1} + \frac{1}{r_3} \]

3. **Substituting the Exradii**:
Substituting the expressions for \( r_1, r_2, r_3 \):
\[ 2 \cdot \frac{s - b}{\Delta} = \frac{s - a}{\Delta} + \frac{s - c}{\Delta} \]

4. **Clearing the Denominator**:
Multiply through by \( \Delta \) (assuming \( \Delta \neq 0 \)):
\[ 2(s - b) = (s - a) + (s - c) \]

5. **Simplifying the Equation**:
Expanding and simplifying:
\[ 2s - 2b = s - a + s - c \]
\[ 2s - 2b = 2s - (a + c) \]

6. **Rearranging the Terms**:
Cancel \( 2s \) from both sides:
\[ -2b = - (a + c) \]
Thus, we have:
\[ 2b = a + c \]

7. **Conclusion**:
The equation \( 2b = a + c \) indicates that the sides \( a, b, c \) are in Arithmetic Progression (AP).

### Final Answer:
If the exradii \( r_1, r_2, r_3 \) of a triangle are in HP, then the sides \( a, b, c \) are in AP.