The incentre of a triangle is determined by the :

Options

A. medians
B. angle bisector
C. perpendicular bisectors
D. altitudes

Answer

To determine the incenter of a triangle, we need to understand the properties of the triangle and the significance of various lines drawn within it. Here’s a step-by-step solution to the question:

### Step-by-Step Solution:

1. **Understanding the Triangle**:
- Let’s denote the triangle as ABC, where A, B, and C are the vertices of the triangle.

2. **Identifying the Options**:
- The options given are:
1. Median
2. Angle Bisector
3. Perpendicular Bisector
4. Altitude

3. **Explaining the Median**:
- A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. It divides the triangle into two smaller triangles of equal area. The point where all three medians intersect is called the centroid.
- **Conclusion**: The median does not determine the incenter.

4. **Explaining the Perpendicular Bisector**:
- A perpendicular bisector of a side of a triangle is a line that is perpendicular to that side and divides it into two equal parts. The point where the perpendicular bisectors of all three sides intersect is called the circumcenter.
- **Conclusion**: The perpendicular bisector does not determine the incenter.

5. **Explaining the Altitude**:
- An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side. The intersection point of the altitudes is known as the orthocenter.
- **Conclusion**: The altitude does not determine the incenter.

6. **Explaining the Angle Bisector**:
- An angle bisector of a triangle is a line that divides an angle into two equal angles. The point where the angle bisectors of all three angles intersect is called the incenter. The incenter is also the center of the circle inscribed within the triangle (incircle).
- **Conclusion**: The incenter is determined by the angle bisector.

7. **Final Answer**:
- Therefore, the incenter of a triangle is determined by the angle bisector.