Incentre of a triangle lies in the interior of :

Options

A. an isosceles triangle only
B. a right angled triangle only
C. any equilateral triangle only
D. any triangle

Answer

To determine where the incenter of a triangle lies, we need to analyze the properties of the incenter in relation to different types of triangles. The incenter is the point where the angle bisectors of a triangle intersect, and it is equidistant from all three sides of the triangle.

### Step-by-Step Solution:

1. **Understanding the Incenter**:
- The incenter of a triangle is defined as the point where the three angle bisectors of the triangle intersect. It is denoted as point 'I'.
- The incenter is always located inside the triangle, regardless of the type of triangle.

2. **Analyzing Different Types of Triangles**:
- **Isosceles Triangle**:
- Draw an isosceles triangle ABC, where AB = AC.
- Construct the angle bisectors from each vertex (A, B, and C).
- The intersection point of these angle bisectors (I) will be inside the triangle.

3. **Right Angle Triangle**:
- Draw a right triangle, say triangle DEF, where angle D is the right angle.
- Again, construct the angle bisectors from each vertex (D, E, and F).
- The incenter (I) will also lie inside this triangle.

4. **Equilateral Triangle**:
- Draw an equilateral triangle GHI, where all sides are equal.
- Construct the angle bisectors from each vertex (G, H, and I).
- The incenter (I) will be at the center of the triangle, thus also inside.

5. **Conclusion**:
- From the analysis of the isosceles triangle, right triangle, and equilateral triangle, we can conclude that the incenter lies within the interior of all types of triangles.
- Therefore, the answer to the question is that the incenter of a triangle lies in the interior of **any triangle**.

### Final Answer:
The incenter of a triangle lies in the interior of **any triangle**.