In any triangle, the three medians are the line segments that join each vertex to the midpoint of the opposite side. The point at which all three medians meet is called the ______ of the triangle.

Introduction / Context:
This is a conceptual question about special points associated with a triangle. Triangles have several important centres, such as the centroid, incentre, circumcentre and orthocentre, each defined in terms of a specific construction. Understanding these definitions is essential for geometry and aptitude exams, because many problems refer to these points or rely on their properties. Here, the definition focuses on medians, which are segments joining each vertex of the triangle to the midpoint of the opposite side. The point where all three medians intersect has a specific name and important properties related to balancing and area.

Given Data / Assumptions:
• A triangle has three medians, one from each vertex to the midpoint of the opposite side.
• The question refers to the common point of intersection of all three medians.
• We are asked to identify the correct name of this point from the given options: centroid, incentre, circumcentre and orthocentre.

Concept / Approach:
The key is to recall the definitions of the four classical triangle centres:
• Centroid: point of intersection of the three medians.
• Incentre: point of intersection of the internal angle bisectors, and centre of the inscribed circle touching all three sides.
• Circumcentre: point of intersection of the perpendicular bisectors of the three sides, and centre of the circumscribed circle passing through all three vertices.
• Orthocentre: point of intersection of the three altitudes of the triangle.
Since the question explicitly mentions medians, we identify that the correct term is centroid.

Step-by-Step Solution:
Step 1: Recall that a median of a triangle is a line segment from a vertex to the midpoint of the opposite side. Step 2: Note that every triangle has exactly three medians, one from each vertex. Step 3: The three medians are always concurrent, meaning they intersect at a single point inside the triangle. Step 4: By definition, the point of intersection of the three medians is called the centroid of the triangle. Step 5: Therefore, the blank in the question should be filled with the word “centroid”.

Verification / Alternative check:
We can cross check by thinking about the other centres. The incentre is related to angle bisectors and equal distances from sides, not medians. The circumcentre is where perpendicular bisectors of sides meet, often outside the triangle for obtuse triangles. The orthocentre is where altitudes meet and may lie inside, on, or outside the triangle. Only the centroid is defined using medians, and it always lies inside the triangle and divides each median in the ratio 2 : 1 from the vertex to the base, which is consistent with the description given in the question.

Why Other Options Are Wrong:
“Incentre” is incorrect here because it is associated with internal angle bisectors and the incircle, not medians. “Circumcentre” is wrong because it relates to perpendicular bisectors and the circumcircle, not segments from vertices to midpoints. “Orthocentre” is the meeting point of altitudes, which are perpendicular from vertices to opposite sides, and does not involve midpoints in its definition. Therefore, none of these match the idea of intersection of medians, leaving “centroid” as the only correct term.

Common Pitfalls:
Students often confuse the four centres because their names sound similar and all refer to intersection points of certain lines in a triangle. A useful memory aid is to link the prefix to its construction: “cen” in centroid can be associated with “centre of mass,” matching the balancing point of medians; “in” in incentre suggests a circle inscribed inside the triangle; “circum” in circumcentre points to a circle drawn around the triangle; and “ortho” in orthocentre relates to orthogonal or perpendicular altitudes. Keeping these associations clear helps avoid mixing up the definitions.

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