In a triangle `cot A cot B cot C gt 0`, then the triangle is

Options

A. acute angled
B. rt. angled
C. obtuse angled
D. none

Answer

To solve the problem, we need to analyze the condition given: \( \cot A \cot B \cot C > 0 \). This condition implies that the product of the cotangents of the angles of triangle \( ABC \) is positive.

### Step-by-Step Solution:

1. **Understanding the Condition**:
- The cotangent function, \( \cot \theta \), is positive when the angle \( \theta \) is in the range \( (0, \frac{\pi}{2}) \). This means that each angle \( A \), \( B \), and \( C \) must be acute angles (less than \( 90^\circ \)).

2. **Analyzing the Angles**:
- Since \( \cot A > 0 \), it implies \( A \) must be in the interval \( (0, \frac{\pi}{2}) \).
- Similarly, \( \cot B > 0 \) implies \( B \) must also be in the interval \( (0, \frac{\pi}{2}) \).
- Lastly, \( \cot C > 0 \) implies \( C \) must be in the interval \( (0, \frac{\pi}{2}) \).

3. **Summing the Angles**:
- The sum of angles in any triangle is \( A + B + C = \pi \).
- If all angles \( A \), \( B \), and \( C \) are less than \( \frac{\pi}{2} \), then their sum \( A + B + C \) must also be less than \( \frac{3\pi}{2} \), which is consistent with the triangle's angle sum property.

4. **Conclusion**:
- Since all angles \( A \), \( B \), and \( C \) are in the interval \( (0, \frac{\pi}{2}) \), we conclude that triangle \( ABC \) is an **acute triangle**.

### Final Answer:
The triangle is **acute**.