Which pair of centres given below lie outside a acute angle triangle?

Options

A. Circumcentre and centroid
B. Incentre and centroid
C. Circumcentre and orthocentre
D. None of these

Answer

To determine which pair of centers lies outside an acute-angled triangle, we need to understand the properties of the centers of a triangle: the circumcenter, incenter, centroid, and orthocenter.

### Step-by-Step Solution:

1. **Identify the Centers**: The four main centers of a triangle are:
- **Circumcenter**: The point where the perpendicular bisectors of the sides intersect.
- **Incenter**: The point where the angle bisectors of the angles intersect.
- **Centroid**: The point where the medians of the triangle intersect.
- **Orthocenter**: The point where the altitudes of the triangle intersect.

2. **Properties of Centers in Acute Triangles**:
- In an **acute triangle**, all angles are less than 90 degrees.
- **Circumcenter**: In an acute triangle, the circumcenter lies inside the triangle.
- **Incenter**: The incenter, being the intersection of the angle bisectors, also lies inside the triangle.
- **Centroid**: The centroid, which is the intersection of the medians, is always inside the triangle.
- **Orthocenter**: The orthocenter, which is the intersection of the altitudes, also lies inside an acute triangle.

3. **Analyze the Options**:
- **Circumcenter and Centroid**: Both lie inside the triangle.
- **Incenter and Centroid**: Both lie inside the triangle.
- **Circumcenter and Orthocenter**: Both lie inside the triangle.
- **None of these**: This implies that no centers lie outside the triangle.

4. **Conclusion**: Since all centers (circumcenter, incenter, centroid, and orthocenter) of an acute triangle lie inside the triangle, the correct answer is **D: None of these**.