Three vertical poles of heights `h_1, h_2 and h_3` at the vertices A, B and C of a `angleABC` subtend angles `alpha,beta `and `gamma` respectively at the cicumcentre of triangle. If `cotalpha, cotbeta` and `cotgamma` are in A.P. then `h_1,h_2,h_3,` are in

Options

A. A.P.
B. G.P.
C. H.P.
D. none of these

Explanation

To solve the problem step by step, we will analyze the given information and derive the necessary relationships.

### Step 1: Understand the Problem
We have three vertical poles of heights \( h_1, h_2, h_3 \) located at the vertices \( A, B, C \) of triangle \( ABC \). These poles subtend angles \( \alpha, \beta, \gamma \) respectively at the circumcenter \( O \) of the triangle. We are given that \( \cot \alpha, \cot \beta, \cot \gamma \) are in arithmetic progression (A.P.). We need to determine the progression in which \( h_1, h_2, h_3 \) lie.

**Hint:** Identify the relationships between the angles and the heights of the poles.

### Step 2: Set Up the Geometry
Let \( R \) be the circumradius of triangle \( ABC \). The heights of the poles can be expressed in terms of the angles and the circumradius:
- \( h_1 = R \tan \alpha \)
- \( h_2 = R \tan \beta \)
- \( h_3 = R \tan \gamma \)

**Hint:** Use the definition of tangent in relation to the angles subtended at the circumcenter.

### Step 3: Relate Cotangent to Tangent
Recall that:
\[ \tan \theta = \frac{1}{\cot \theta} \]
Thus, we can express the heights as:
- \( h_1 = \frac{R}{\cot \alpha} \)
- \( h_2 = \frac{R}{\cot \beta} \)
- \( h_3 = \frac{R}{\cot \gamma} \)

**Hint:** Rewrite the heights in terms of cotangent to see the relationship more clearly.

### Step 4: Analyze the Given Condition
Since \( \cot \alpha, \cot \beta, \cot \gamma \) are in A.P., we can express this condition mathematically:
\[ 2 \cot \beta = \cot \alpha + \cot \gamma \]

**Hint:** Use the property of arithmetic progression to set up the equation.

### Step 5: Formulate the Heights in Terms of A.P.
From the relationship of cotangents, we can derive:
\[ \frac{1}{h_1} + \frac{1}{h_3} = 2 \cdot \frac{1}{h_2} \]
This implies that:
\[ \frac{h_2}{h_1} + \frac{h_2}{h_3} = 2 \]
or equivalently,
\[ h_1, h_2, h_3 \text{ are in Harmonic Progression (H.P.)} \]

**Hint:** Recognize that the reciprocal of the heights being in arithmetic progression indicates that the heights themselves are in harmonic progression.

### Conclusion
Thus, we conclude that if \( \cot \alpha, \cot \beta, \cot \gamma \) are in A.P., then \( h_1, h_2, h_3 \) are in Harmonic Progression (H.P.).

**Final Answer:** \( h_1, h_2, h_3 \) are in Harmonic Progression (H.P.).