Centroid of a Triangle

The centroid of a triangle is the point where the three medians of the triangle intersect. It is often called the centre of mass or the balance point of the triangle.

        A
       / \
      /  O\   O = centroid
     /      \
    B--------C

No matter what type of triangle it is—scalene, isosceles, or obtuse—the centroid always lies inside the triangle.

Definition: The centroid of a triangle is the common point where all three medians intersect.

If medians from vertices \(A\), \(B\), and \(C\) meet at point \(G\), then \(G\) is the centroid of \(\triangle ABC\).

• A triangle's three medians always meet at one single point.
• The centroid divides each median in the ratio 2:1.
• The longer segment is always from the vertex to the centroid.
• The centroid is the triangle’s balancing point.
• The centroid always lies inside the triangle.

If the vertices of a triangle have coordinates:

\(A(x_1, y_1),\ B(x_2, y_2),\ C(x_3, y_3)\)

then the centroid \(G\) is given by the formula:

\( G\left(\dfrac{x_1 + x_2 + x_3}{3},\ \dfrac{y_1 + y_2 + y_3}{3}\right) \)

This makes it easy to locate the centroid on graphs or maps.

To construct the centroid of \(\triangle ABC\):

1. Find the midpoint of side \(BC\). Connect it to vertex \(A\) → this is one median.
2. Find the midpoint of side \(AC\). Connect it to vertex \(B\) → second median.
3. The point where these medians meet is the centroid \(G\).
4. (The third median will also pass through the same point.)

The centroid plays a major role in:

• designing stable triangular structures,
• balancing physical objects,
• coordinate geometry calculations,
• dividing shapes into equal-area parts,
• applications in physics (centre of mass and equilibrium).

It is one of the most used points in triangle geometry.