Definition: The centroid of a triangle is the point where its three medians meet, the triangle center of mass, two thirds of the way along each median from a vertex.
What is a centroid?
A median joins a vertex to the midpoint of the opposite side. The three medians always cross at one point, the centroid. It is always inside the triangle, and a cardboard triangle balances on a pencil tip placed there.
What is the 2:1 rule?
The centroid divides each median so that the part from the vertex is twice as long as the part to the midpoint. In coordinates, average the three vertices: ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3).
Example. A(2, 9), B(8, 3), C(-4, 6). The x-average is (2 + 8 – 4)/3 = 2. The y-average is (9 + 3 + 6)/3 = 6. The centroid is (2, 6). If a median has total length 24, the vertex-to-centroid part is 16 and the rest is 8.
What errors are frequent?
Confusing the median with an altitude, which is perpendicular to the side and does not need to hit the midpoint. Another error is dividing by 2 as for a midpoint, when three points require division by 3.