In a triangle, if orthocentre, circumcentre, incentre and centroid coincide, then the triangle must be

Options

A. obtuse angled
B. isosceles
C. equilateral
D. right - angled

Answer

To solve the problem, we need to determine the type of triangle in which the orthocenter, circumcenter, incenter, and centroid all coincide at a single point.

### Step-by-Step Solution:

1. **Understanding the Points of Concurrency**:
- The orthocenter (H) is the point where the three altitudes of a triangle intersect.
- The circumcenter (O) is the point where the perpendicular bisectors of the sides intersect, and it is the center of the circumcircle.
- The incenter (I) is the point where the angle bisectors of the triangle intersect, and it is the center of the incircle.
- The centroid (G) is the point where the three medians of the triangle intersect.

2. **Condition for Coincidence**:
- For all four points (H, O, I, G) to coincide, the triangle must have special properties. In general, these points are distinct for most triangles.

3. **Analyzing Special Triangles**:
- In an **equilateral triangle**, all sides and angles are equal. This symmetry causes the orthocenter, circumcenter, incenter, and centroid to all be located at the same point.

4. **Conclusion**:
- Therefore, if the orthocenter, circumcenter, incenter, and centroid of a triangle coincide, the triangle must be an **equilateral triangle**.

### Final Answer:
The triangle must be an **equilateral triangle**.