The triangle in which the orthocenter, circumcentre, Incenter and the centroid coincide each other at a point is known as ___________.

Options

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

Answer

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

### Step-by-Step Solution:

1. **Understanding Key Points in a Triangle**:
- **Orthocenter**: The point where the three altitudes of a triangle intersect.
- **Circumcenter**: The point where the perpendicular bisectors of the sides of a triangle intersect.
- **Incenter**: The point where the angle bisectors of a triangle intersect.
- **Centroid**: The point where the three medians of a triangle intersect.

2. **Analyzing the Coincidence of Points**:
- In a general triangle, these four points do not coincide. They are distinct and have specific properties related to the triangle's shape and angles.
- However, there is a special case where all these points coincide at one point.

3. **Identifying the Special Triangle**:
- The only type of triangle where the orthocenter, circumcenter, incenter, and centroid all coincide is the **equilateral triangle**.
- In an equilateral triangle, all sides and angles are equal, which leads to the coincidence of these points.

4. **Conclusion**:
- Therefore, the triangle in which the orthocenter, circumcenter, incenter, and centroid coincide at a point is known as an **equilateral triangle**.

### Final Answer:
The triangle in which the orthocenter, circumcenter, incenter, and centroid coincide at a point is known as an **equilateral triangle**.

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