The point of intersection of all the internal angle bisectors of a triangle is called the ______ of the triangle.

Introduction / Context:
This conceptual geometry question checks basic knowledge of special points associated with a triangle. Every triangle has several important interior points defined by different constructions, such as midpoints, perpendiculars and angle bisectors. The question specifically refers to the point where all three internal angle bisectors meet. Knowing the names and properties of these centres is essential in many coordinate geometry, construction and problem solving situations.

Given Data / Assumptions:
- We are dealing with a general triangle, with three interior angles. - The question considers the internal bisectors of each of these three angles. - All three angle bisectors are extended until they intersect at a single common point. - We must recall the standard name for this point.

Concept / Approach:
A triangle has four classical centres that are typically introduced in school level geometry. The centroid is where the three medians intersect. The circumcenter is where the perpendicular bisectors of the sides meet. The orthocenter is where the three altitudes intersect. Finally, the incenter is where the three internal angle bisectors meet. The incenter has a special property: it is the centre of the circle that can be inscribed inside the triangle, touching all three sides. By matching the description in the question to the correct definition, we identify the right centre.

Step-by-Step Solution:
Step 1: Identify which lines are being used: the lines mentioned are the three internal angle bisectors of the triangle. Step 2: Recall that an angle bisector divides an angle into two equal parts. Step 3: In any triangle, the three internal angle bisectors are concurrent, meaning they all meet at a single point. Step 4: The standard name for this point of concurrency is the incenter. Step 5: Therefore, the point where the three internal angle bisectors meet is called the incenter of the triangle. Step 6: The incenter is also the centre of the inscribed circle, which is the unique circle tangent to all three sides of the triangle.

Verification / Alternative check:
To verify, we can compare the other listed centres. The circumcenter is defined as the intersection of perpendicular bisectors of the sides and is the centre of the circumscribed circle passing through the vertices. The centroid is the intersection of the medians, each joining a vertex to the midpoint of the opposite side, and it represents the balancing point of the triangle. The orthocenter is where the three altitudes meet, with each altitude being a perpendicular from a vertex to the opposite side. None of these definitions involve the intersection of angle bisectors, so they cannot match the statement in the question. This cross check confirms that only the incenter fits.

Why Other Options Are Wrong:
Option Circumcenter involves perpendicular bisectors of sides, not internal angle bisectors. Option Centroid involves medians, not angle bisectors, so it is unrelated to the given description. Option Orthocenter is the intersection of altitudes, which are perpendiculars from vertices to opposite sides, again unrelated to angle bisectors.

Common Pitfalls:
Students sometimes confuse the various centres because they all involve three lines intersecting at a point. A reliable way to remember is to link each name to a keyword: incenter with inscribed circle and angle bisectors, circumcenter with circumscribed circle and perpendicular bisectors, centroid with centre of mass and medians, and orthocenter with orthogonal altitudes. Writing down these associations on a formula sheet can help avoid mixing them up during exams.

Final Answer:
The point of intersection of all the internal angle bisectors of a triangle is called the Incenter of the triangle.