The point inside a triangle that is equidistant from all three sides, and is the common point of intersection of the internal angle bisectors, is called the ______ of the triangle.

Introduction / Context:
This conceptual question is about identifying one of the four classical centres of a triangle: the incentre, circumcentre, centroid and orthocentre. Each of these is defined in terms of a specific type of line drawn inside the triangle. The point being described here is equidistant from all three sides of the triangle and is formed by the intersection of the internal angle bisectors. Recognising which centre has this property is important in many geometry and aptitude problems where distances to sides or inscribed circles are involved.

Given Data / Assumptions:
• The point is equidistant from all three sides of the triangle.
• It is the common point of intersection of all the internal angle bisectors of the triangle.
• The problem asks for the correct name of this point among the standard triangle centres.

Concept / Approach:
We recall the definitions of the main triangle centres:
• Incentre: intersection of the internal angle bisectors; centre of the inscribed circle (incircle) that touches all three sides; equidistant from all sides.
• Circumcentre: intersection of the perpendicular bisectors of the sides; centre of the circumcircle passing through all three vertices; equidistant from all vertices, not sides.
• Orthocentre: intersection of the three altitudes; not defined in terms of equal distances to sides or vertices in this way.
• Centroid: intersection of the three medians; centre of mass but not equidistant from sides or vertices. Since the point described is equidistant from the sides and is formed by angle bisectors, it matches the definition of the incentre.

Step-by-Step Solution:
Step 1: Identify that the point is obtained by intersecting the internal angle bisectors of the triangle. Step 2: Recall that the unique point where the three internal angle bisectors meet is called the incentre of the triangle. Step 3: Note that the incentre is the centre of the incircle, which is a circle drawn inside the triangle and tangent to all three sides. Step 4: Because a circle is tangent to a line at exactly one point and has equal radius to each tangency point, the incentre is equidistant from all three sides. Step 5: Therefore, the blank should be filled with the term “Incentre”.

Verification / Alternative check:
Compare this with the other triangle centres. The circumcentre is equidistant from the vertices, not from the sides, so it cannot be the point described. The centroid is defined via medians, not angle bisectors, and does not maintain equal distance from either sides or vertices. The orthocentre is the point where altitudes meet and does not generally have any equal distance property like this. Only the incentre simultaneously satisfies being the intersection of internal angle bisectors and being equidistant from all sides because it is the centre of the inscribed circle that touches each side at exactly one point.

Why Other Options Are Wrong:
“Circumcenter” is incorrect here because it is tied to perpendicular bisectors of sides and equal distances to vertices, which is a different concept. “Orthocentre” is based on altitudes and does not guarantee equal distances to sides or vertices. “Centroid” is based on medians and represents the balance point of the triangle but does not maintain equal distances from the sides. Since the question clearly emphasises equal distance from sides and intersection of angle bisectors, only “Incentre” fits both criteria.

Common Pitfalls:
Many learners confuse the triangle centres because their names sound similar and they are often introduced together. A useful way to remember is to link words to their roles: “in” in incentre suggests a circle inscribed inside the triangle, “circum” in circumcentre suggests a circle around the triangle, “ortho” relates to perpendicular altitudes, and “cen” in centroid evokes the centre of mass via medians. Keeping these associations clear helps you quickly select the correct centre when a question describes specific geometric properties.

Final Answer:
The point equidistant from the sides of a triangle is called the Incentre of the triangle.