Triangle medians & the centroid

Find the balance point of a triangle.

The idea

A median joins a vertex to the midpoint of the opposite side. All three medians meet at the centroid G, which lies inside every non-degenerate triangle. The centroid divides each median in a 2:1 ratio, with the longer part next to the vertex.

Work through an example

For A(−4, −3), B(4, −3), and C(0, 4), the centroid is (0, −2/3). The base AB has length 8 and the height from C is 7, so the area is 28 square units. Moving C horizontally while keeping its height preserves that area.

What to watch for

A median is not necessarily perpendicular to its opposite side. The centroid, circumcenter, incenter, and orthocenter are different points in general; they coincide in an equilateral triangle.