To solve the problem step by step, we need to establish that if \( \sin A, \sin B, \sin C \) are in arithmetic progression (A.P.), then the altitudes from vertices A, B, and C of triangle ABC will be in harmonic progression (H.P.).
### Step-by-Step Solution:
1. **Understanding the Given Condition**:
- We are given that \( \sin A, \sin B, \sin C \) are in A.P. This means:
\[
2 \sin B = \sin A + \sin C
\]
2. **Using the Area of the Triangle**:
- The area \( \Delta \) of triangle ABC can be expressed in terms of its sides and corresponding altitudes:
\[
\Delta = \frac{1}{2} \times a \times P_1 = \frac{1}{2} \times b \times P_2 = \frac{1}{2} \times c \times P_3
\]
- Here, \( P_1, P_2, P_3 \) are the altitudes from vertices A, B, and C respectively.
3. **Expressing Altitudes in Terms of Area**:
- From the area expressions, we can derive the altitudes:
\[
P_1 = \frac{2\Delta}{a}, \quad P_2 = \frac{2\Delta}{b}, \quad P_3 = \frac{2\Delta}{c}
\]
4. **Using the Sine Rule**:
- According to the sine rule:
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = k \quad \text{(some constant)}
\]
- Thus, we can express the sides in terms of the sine values:
\[
a = k \sin A, \quad b = k \sin B, \quad c = k \sin C
\]
5. **Substituting into Altitude Formulas**:
- Substitute \( a, b, c \) in the altitude formulas:
\[
P_1 = \frac{2\Delta}{k \sin A}, \quad P_2 = \frac{2\Delta}{k \sin B}, \quad P_3 = \frac{2\Delta}{k \sin C}
\]
6. **Relating the Altitudes**:
- We can express the altitudes as:
\[
P_1 = \frac{2\Delta}{k \sin A}, \quad P_2 = \frac{2\Delta}{k \sin B}, \quad P_3 = \frac{2\Delta}{k \sin C}
\]
7. **Finding the Relationship**:
- Since \( \sin A, \sin B, \sin C \) are in A.P., we know that:
\[
\frac{1}{\sin A}, \frac{1}{\sin B}, \frac{1}{\sin C}
\]
- will be in H.P. (Harmonic Progression).
8. **Conclusion**:
- Therefore, since the reciprocals of the sines are in H.P., it follows that the altitudes \( P_1, P_2, P_3 \) are also in H.P.
### Final Answer:
The altitudes \( P_1, P_2, P_3 \) from vertices A, B, and C of triangle ABC are in harmonic progression (H.P.).
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