Triangle geometry/Median/Introduction/Section

Definition

For a set of $n$ points $P_1,\dots ,P_n$ in an affine space $E$ over a real vector space $V$, the barycentric combination

$\frac{1}{n}{P}_1+\frac{1}{n}{P}_2+\cdots +\frac{1}{n}{P}_n$

is called the center of gravity

of these points.

This is also called the centroid of the points. It just the barycentric combination of the points where every point enters with the same weight. For two points $P,Q\in E$, the center of gravity $\frac{1}{2}P+\frac{1}{2}Q$ is also called the midpoint of the two points (or of the line segment $\overline{P,Q}$). In the case of two real numbers, we also talk about the arithmetic mean of the numbers. For $n=1,2,3$, the centroid of the points is also the centroid of its convex hull. The centroid of the three points is also the intersection point of the medians of the triangle.

Definition

For a nondegenerate triangle $A,B,C$ in a Euclidean plane, the line

$\frac{B+C}{2}+s(A-\frac{B+C}{2}),s\in ℝ,$

is called the median

through $A$.

The median through $A$ runs through the point $A$ and the midpoint of the opposite side of the triangle.

Theorem

In a nondegenerate triangle in the Euclidean plane, the three medians

intersect in the centroid $\frac{A+B+C}{3}$ of the triangle.

Proof

We have

$\frac{B+C}{2}+\frac{1}{3}(A-\frac{B+C}{2})=\frac{1}{3}A+(\frac{1}{2}-\frac{1}{6})B+(\frac{1}{2}-\frac{1}{6})C=\frac{1}{3}A+\frac{1}{3}B+\frac{1}{3}C,$

and this is a point of the median through $A$. Due to the symmetry of the situation, this point also lies on the other medians.

$◻$

In particular, the centroid divides every median with the ratio $2:1$, the longer part being closer to the vertex.