Answer :
To solve this problem, we need to determine which geometric term correctly describes the point where the three angle bisectors of a triangle intersect.
### Key Terms:
1. Circumcenter: The point where the perpendicular bisectors of a triangle intersect. It is equidistant from all three vertices of the triangle.
2. Centroid: The point where the three medians of a triangle intersect. It is also the center of mass or the balance point of the triangle.
3. Incenter: The point where the three angle bisectors of a triangle intersect. It is equidistant from all three sides of the triangle.
4. Orthocenter: The point where the three altitudes of a triangle intersect.
### Explanation:
An angle bisector of a triangle is a line segment that divides an angle of the triangle into two equal parts. The incenter is the unique point where the angle bisectors of all three angles of a triangle meet or intersect. This point is significant because it is the center of the triangle's inscribed circle (incircle), which is the largest circle that fits inside the triangle and touches all three sides.
Given this understanding, the correct term that describes the point where the three angle bisectors of a triangle intersect is Incenter.
### Conclusion:
The point where the three angle bisectors of a triangle intersect is called the Incenter.
So, the correct answer is:
C. Incenter.
### Key Terms:
1. Circumcenter: The point where the perpendicular bisectors of a triangle intersect. It is equidistant from all three vertices of the triangle.
2. Centroid: The point where the three medians of a triangle intersect. It is also the center of mass or the balance point of the triangle.
3. Incenter: The point where the three angle bisectors of a triangle intersect. It is equidistant from all three sides of the triangle.
4. Orthocenter: The point where the three altitudes of a triangle intersect.
### Explanation:
An angle bisector of a triangle is a line segment that divides an angle of the triangle into two equal parts. The incenter is the unique point where the angle bisectors of all three angles of a triangle meet or intersect. This point is significant because it is the center of the triangle's inscribed circle (incircle), which is the largest circle that fits inside the triangle and touches all three sides.
Given this understanding, the correct term that describes the point where the three angle bisectors of a triangle intersect is Incenter.
### Conclusion:
The point where the three angle bisectors of a triangle intersect is called the Incenter.
So, the correct answer is:
C. Incenter.
