GeoProof — Centroid & the 2:1 Median explorer

AG : GMₐ = BG : GM_b = CG : GM_c = 2 : 1

The Centroid & the 2:1 Median

A median of a triangle joins a vertex to the midpoint of the opposite side. A triangle has three of them, and they always pass through a single point — the centroid G. The centroid cuts each median in the same ratio:

AG : GMₐ = 2 : 1

That is, the distance from the vertex to G is exactly twice the distance from G to the midpoint, for all three medians at once.

The centroid is the average point

In coordinates, G is just the average of the vertices: G = (A + B + C) / 3. It is the triangle's center of mass — balance a triangular plate on a pin at G and it stays level.

Six triangles of equal area

The three medians slice the triangle into six smaller triangles. Although they look different, all six have exactly the same area — one-sixth of the whole. Turn on "Equal-area regions" to see them shaded.

Why 2:1?

Connect the midpoints of two sides: that midsegment is parallel to the third side and half its length. The little triangle it forms with the centroid is similar to the big one at a 1:2 scale, which forces the centroid to sit two-thirds of the way down each median — the 2:1 split.

A bit of history

Archimedes (c. 287–212 BCE) studied centroids in his work on balance and centers of gravity; the 2:1 median property follows from that classical theory of the lever.