Each side of `DeltaABC` is the polarof the opposite vertex with respect to a circle with centre P. For the `DeltaABC` the point P is

Options

A. centroid
B. circum centre
C. incentre
D. ortho centre

Answer

To solve the problem, we need to show that the point \( P \) is the circumcenter of triangle \( \Delta ABC \) when each side of the triangle is the polar of the opposite vertex with respect to a circle centered at \( P \).

### Step-by-Step Solution:

1. **Understanding the Concept of Polar and Vertex**:
- The polar of a point with respect to a circle is the line that represents all points from which the tangents to the circle from that point are equal in length. In this case, each side of triangle \( \Delta ABC \) is the polar of the opposite vertex.

2. **Drawing the Triangle and Circle**:
- Draw triangle \( \Delta ABC \) with vertices \( A \), \( B \), and \( C \).
- Mark point \( P \) as the center of the circle. Draw the circle with center \( P \) that intersects the triangle.

3. **Identifying the Polars**:
- Identify the sides of the triangle:
- Side \( BC \) is the polar of vertex \( A \).
- Side \( CA \) is the polar of vertex \( B \).
- Side \( AB \) is the polar of vertex \( C \).

4. **Using the Property of Polars**:
- Since each side of the triangle is the polar of the opposite vertex, we can conclude that the distances from point \( P \) to each side of the triangle are equal. This is a property of the circumcenter.

5. **Conclusion**:
- Since \( P \) is equidistant from all three sides of triangle \( \Delta ABC \), it follows that \( P \) is the circumcenter of the triangle. Therefore, the point \( P \) is the circumcenter of triangle \( \Delta ABC \).

### Final Statement:
Thus, we conclude that the point \( P \) is the circumcenter of triangle \( \Delta ABC \).

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