Which groups of centres of a triangle given below always lie in a straight line (i.e., centres are collinear)

Options

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

Answer

To determine which groups of centers of a triangle are always collinear, we need to analyze the definitions and relationships between the centers of a triangle: the centroid, orthocenter, and circumcenter.

### Step-by-Step Solution:

1. **Understand the Centers of a Triangle**:
- **Centroid (G)**: The point where the three medians of the triangle intersect. It is the average of the vertices' coordinates.
- **Orthocenter (H)**: The point where the three altitudes of the triangle intersect.
- **Circumcenter (O)**: The point where the three perpendicular bisectors of the sides of the triangle intersect.

2. **Identify the Relationship**:
- It is a known geometric property that the centroid, orthocenter, and circumcenter of a triangle are always collinear. This line is known as the **Euler line**.

3. **Ratio Division**:
- The centroid divides the segment joining the orthocenter and circumcenter in a specific ratio of 2:1. This means that if you were to draw a line segment from the orthocenter to the circumcenter, the centroid would be located two-thirds of the way from the orthocenter to the circumcenter.

4. **Conclusion**:
- Since the centroid, orthocenter, and circumcenter are always collinear, the correct answer to the question is that the group consisting of these three centers is always in a straight line.

### Answer:
The group of centers that are always collinear is: **Centroid, Orthocenter, and Circumcenter**.