What is the centroid?

How the balance point is found

The centroid is always inside the triangle. Along each median, it lies two-thirds of the way from the vertex to the midpoint of the opposite side, so it divides every median in a two-to-one ratio.

This point matters because it links segment structure to geometric meaning. Once the medians are known, the centroid gives a natural answer to balance, subdivision, and coordinate questions.

Along a median, imagine placing a tiny weight at the triangle's vertices. The centroid is the point where the flat triangular plate would balance. The two-to-one division is not just a memorised ratio; it tells you how close the balance point sits to the vertex compared with the opposite midpoint.

If the vertices have coordinates, average their x-values and average their y-values to find the centroid. The coordinate method agrees with the three-median picture, giving a useful check between algebra and geometry.

• The centroid is the point where the medians intersect.
• The centroid is formed by the intersection of the three medians.
• It divides each median in a 2:1 ratio, with the longer part nearer the vertex.
• The centroid is always inside the triangle.

Where the centroid helps

• Use the centroid in median problems, coordinate geometry, and balance-point reasoning.
• Use it when a proof or calculation depends on the intersection of medians.
• Use it to understand how a triangle can be partitioned into equal-area regions by medians.

Three centroid checks

• Do not confuse the centroid with the incenter, circumcenter, or orthocenter; each comes from a different family of lines.
• Do not forget the 2:1 median ratio when measuring from a vertex to the centroid.
• Do not place the centroid by eye alone; it is determined by the medians, not by a guessed middle point.