the centroid's 2:1 median ratio

Every triangle has a single balance point called the centroid — the spot where a cardboard triangle would balance perfectly on a pin. It is where the three medians (each median joins a vertex to the midpoint of the opposite side) all cross. The remarkable fact is not just that the three medians meet at one point, but exactly where on each median that point sits: it cuts every median in the ratio 2 : 1.

Precisely, the centroid G lies on each median two-thirds of the way from the vertex to the midpoint of the opposite side. So if a median runs from vertex A to the midpoint M of the opposite side, then AG : GM = 2 : 1 — the centroid is twice as far from the vertex as it is from the side. This means AG = (2/3)·AM and GM = (1/3)·AM. In coordinates the centroid is dead simple: its position is the average of the three vertices, G = ((x_1 + x_2 + x_3)/3, (y_1 + y_2 + y_3)/3), which makes the 2 : 1 split easy to verify directly.

This is proportion sitting at the heart of a triangle, and it has real teeth: it is why the centroid is the centre of mass of a uniform triangular plate, and the 2 : 1 ratio appears in physics, engineering, and computer graphics whenever a triangle's balance point is needed. A frequent confusion is to mix the centroid up with the triangle's other centres — the circumcentre (equidistant from the vertices), the incentre (equidistant from the sides), and the orthocentre (where the altitudes meet) — which generally sit at different points. Only the centroid is the median-crossing balance point with the clean 2 : 1 rule.