If in a `/_\ABC, a,b,c` are in A.P. and `P_(1),P_(2),P_(3)` are th altitude from the vertices `A,B` and `C`respectively then

Options

A. `P_(1),P_(2),P_(3)` are in A.P.
B. `P_(1),P_(2),P_(3)` are in H.P.
C. `P_(1)+P_(2)+P_(3)le(3R)/(/_\)`
D. `1/(P_(1))+1/(P_(2))+1/(P_(3))le(3R)/(/_\)`

Explanation

To solve the problem, we will follow a step-by-step approach to derive the relationships between the altitudes \( P_1, P_2, P_3 \) and the sides \( a, b, c \) of triangle \( ABC \) where \( a, b, c \) are in arithmetic progression (A.P.).

### Step 1: Understand the Area of the Triangle
The area \( \Delta \) of triangle \( ABC \) can be expressed using the base and height (altitude) from each vertex:
\[ \Delta = \frac{1}{2} a P_1 = \frac{1}{2} b P_2 = \frac{1}{2} c P_3 \]

### Step 2: Express the Altitudes in Terms of Area
From the area expressions, we can isolate the altitudes:
\[ P_1 = \frac{2\Delta}{a}, \quad P_2 = \frac{2\Delta}{b}, \quad P_3 = \frac{2\Delta}{c} \]

### Step 3: Identify the Relationship between \( a, b, c \)
Since \( a, b, c \) are in A.P., we can express them as:
\[ b = \frac{a + c}{2} \]

### Step 4: Show that \( P_1, P_2, P_3 \) are in Harmonic Progression (H.P.)
To show that \( P_1, P_2, P_3 \) are in H.P., we need to show that:
\[ \frac{1}{P_1}, \frac{1}{P_2}, \frac{1}{P_3} \]
are in A.P.

Calculating the reciprocals:
\[ \frac{1}{P_1} = \frac{a}{2\Delta}, \quad \frac{1}{P_2} = \frac{b}{2\Delta}, \quad \frac{1}{P_3} = \frac{c}{2\Delta} \]

### Step 5: Establish the A.P. Condition
For \( \frac{1}{P_1}, \frac{1}{P_2}, \frac{1}{P_3} \) to be in A.P., the following must hold:
\[ 2 \cdot \frac{b}{2\Delta} = \frac{a}{2\Delta} + \frac{c}{2\Delta} \]
This simplifies to:
\[ b = \frac{a + c}{2} \]
which is true since \( a, b, c \) are in A.P.

### Step 6: Conclusion
Since \( P_1, P_2, P_3 \) are in H.P., we conclude that:
- \( P_1, P_2, P_3 \) are in H.P. if \( a, b, c \) are in A.P.