In `DeltaABC`, sinA, sinB and sinC are in AP, then ex-radii `r_(1),r_(2)` and `r_(3)` will be in

Options

A. AP
B. GP
C. HP
D. None of these

Explanation

To solve the problem, we need to establish the relationship between the ex-radii \( r_1, r_2, \) and \( r_3 \) of triangle \( ABC \) given that \( \sin A, \sin B, \sin C \) are in arithmetic progression (AP).

### Step-by-Step Solution:

1. **Understanding the Condition**:
Since \( \sin A, \sin B, \sin C \) are in AP, we can express this condition mathematically:
\[ 2 \sin B = \sin A + \sin C \]

2. **Using the Sine Rule**:
According to the sine rule in triangle \( ABC \):
\[ \frac{A}{\sin A} = \frac{B}{\sin B} = \frac{C}{\sin C} = k \quad (\text{where } k \text{ is a constant}) \]
Therefore, we can express \( \sin A, \sin B, \sin C \) in terms of \( A, B, C \):
\[ \sin A = \frac{A}{k}, \quad \sin B = \frac{B}{k}, \quad \sin C = \frac{C}{k} \]

3. **Substituting into the AP Condition**:
Substitute the expressions for \( \sin A, \sin B, \sin C \) into the AP condition:
\[ 2 \left(\frac{B}{k}\right) = \frac{A}{k} + \frac{C}{k} \]
Simplifying gives:
\[ 2B = A + C \]

4. **Finding the Ex-radii**:
The ex-radii \( r_1, r_2, r_3 \) are given by:
\[ r_1 = \frac{\Delta}{s - A}, \quad r_2 = \frac{\Delta}{s - B}, \quad r_3 = \frac{\Delta}{s - C} \]
where \( \Delta \) is the area of the triangle and \( s \) is the semi-perimeter:
\[ s = \frac{A + B + C}{2} \]

5. **Expressing \( s - A, s - B, s - C \)**:
From the semi-perimeter \( s \):
\[ s - A = \frac{B + C - A}{2}, \quad s - B = \frac{A + C - B}{2}, \quad s - C = \frac{A + B - C}{2} \]

6. **Finding the Relationship Between Ex-radii**:
Now we can express \( \frac{1}{r_1}, \frac{1}{r_2}, \frac{1}{r_3} \):
\[ \frac{1}{r_1} = \frac{s - A}{\Delta}, \quad \frac{1}{r_2} = \frac{s - B}{\Delta}, \quad \frac{1}{r_3} = \frac{s - C}{\Delta} \]

7. **Substituting the Values**:
Substitute \( s - A, s - B, s - C \) into the equations:
\[ \frac{1}{r_1} + \frac{1}{r_3} = \frac{(B + C - A) + (A + B - C)}{2\Delta} = \frac{2B}{2\Delta} = \frac{B}{\Delta} \]
Thus, we find:
\[ \frac{1}{r_1} + \frac{1}{r_3} = \frac{B}{\Delta} \]

8. **Conclusion**:
From the above relationships, we can conclude that:
\[ \frac{1}{r_1}, \frac{1}{r_2}, \frac{1}{r_3} \text{ are in Harmonic Progression (HP)} \]
Therefore, \( r_1, r_2, r_3 \) are in Harmonic Progression.