A pole stands vertically ,inside a scalene triangular park ABC .If the angle of elevation of the top of the pole from each corner of tha park is same ,then in `DeltaABC` , the foot of the pole is at the

Options

A. centroid
B. circumcentre
C. incentre
D. orthocentre

Answer

To solve the problem, we need to determine the position of the foot of the pole inside the scalene triangular park ABC, given that the angle of elevation from each corner of the park to the top of the pole is the same.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a scalene triangle ABC, and a pole standing vertically inside it.
- The angle of elevation to the top of the pole from each corner (A, B, and C) is the same.

2. **Identifying the Geometry**:
- Let the foot of the pole be point D inside triangle ABC.
- Since the angle of elevation from points A, B, and C to point D is the same, we can denote this angle as θ.

3. **Using Properties of Angles**:
- The fact that the angle of elevation is the same from all three corners implies that the horizontal distances from points A, B, and C to point D must be equal.
- This is because for the same angle of elevation, the height of the pole (the vertical distance from D to the top of the pole) remains constant.

4. **Conclusion about the Position of D**:
- The point D, where the foot of the pole is located, must be equidistant from the three sides of triangle ABC.
- In triangle geometry, the point that is equidistant from all three sides of a triangle is known as the **incenter**.

5. **Final Answer**:
- Therefore, the foot of the pole is at the **incenter** of triangle ABC.