In a triangle, the point of intersection of all three medians is called which special point of the triangle?

Introduction / Context:
This is a conceptual question about special points in a triangle. A triangle has several important points associated with it, including the centroid, orthocentre, circumcentre, and incentre. Each is defined by a different type of line set, such as medians, altitudes, perpendicular bisectors, or angle bisectors. Here, you must recall which special point is formed by the intersection of the three medians of a triangle.

Given Data / Assumptions:

• We are dealing with a general triangle, not restricted to being right, isosceles, or equilateral.
• A median of a triangle is a line segment joining a vertex to the mid point of the opposite side.
• There are three medians, one from each vertex.
• These three medians intersect at a single point.
• We must identify the name of this common point of intersection.

Concept / Approach:
By definition, the centroid of a triangle is the point where all three medians intersect. This point has several key properties: it always lies inside the triangle, it divides each median in the ratio 2 : 1 from vertex to midpoint, and it can be interpreted as the centre of mass of a triangle made from uniform material. Other special points, like the orthocentre, circumcentre, and incentre, are defined using different sets of lines and should not be confused with the centroid.

Step-by-Step Solution:
Recall that a median connects a vertex to the mid point of the opposite side.There are three such medians in a triangle.All three medians are concurrent; they meet at a single point.This common point of intersection is called the centroid of the triangle.Hence, the required special point is the centroid.

Verification / Alternative check:
To further confirm, recall the definitions of the other special points. The orthocentre is the intersection of the three altitudes of a triangle. The circumcentre is the intersection of the perpendicular bisectors of the sides and is the centre of the circumcircle. The incentre is the intersection of the angle bisectors and is the centre of the incircle. None of these mention medians. Only the centroid is defined in terms of medians, so this matches the requirement exactly.

Why Other Options Are Wrong:
Orthocentre is wrong because it involves altitudes, not medians. Incentre is incorrect because it is formed by angle bisectors, not medians. Circumcentre refers to perpendicular bisectors and may lie inside or outside the triangle depending on the type of triangle. Excentre is formed by intersection of internal and external angle bisectors and is not associated with medians at all. Therefore, these options do not match the definition given in the question.

Common Pitfalls:
Learners often confuse these special centres because their names sound similar. Another common issue is mixing up altitudes and medians, since both go from a vertex to the opposite side but in different ways. Remember that medians go to the mid point, altitudes go perpendicular to the side, perpendicular bisectors start from the side and go outward, and angle bisectors split angles. Keeping the definitions straight helps in quickly identifying the centroid as the intersection of medians.

Final Answer:
The point where all three medians of a triangle intersect is called the centroid.