Cut a triangle out of card and it balances on exactly one point — the centroid, the average of its three corners. Draw a line from each corner to the middle of the opposite side and all three cross there, each cut in the same 2-to-1 ratio. Top sits at the center. Slide a corner and watch the balance point follow.
The centroid of a triangle is the arithmetic mean of its vertices, G = (A+B+C)/3. It is where the three MEDIANS (each joining a vertex to the midpoint of the opposite side) meet, and it divides every median in a 2:1 ratio measured from the vertex. It is also the triangle’s centre of mass — the single point it balances on. A fail-loud self-check throws unless G equals (A+B+C)/3 AND lies exactly two-thirds of the way from each vertex to the opposite midpoint.
The centroid is one of several triangle centres (circumcenter, incenter, orthocenter differ); for a UNIFORM lamina it is the true balance point, but a weighted triangle balances elsewhere. The averaging and the 2:1 split are exact.