---
tag: 0ZET0ZH16W
kind: definition
name: Median of a triangle
msc: [51M04]
defines: 0XW8HX8VEV
uses: [0CTNFGYMM2, 0S706JBZBG]
sources:
  - url: https://en.wikipedia.org/wiki/Median_(geometry)
    locator: lead section
---

## Definition

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean space $\mathbb{R}^n$. Let $M_A$ be the [[midpoint|0S706JBZBG]] of $B$ and $C$.
The median from $A$ is the segment $\{\, A + t (M_A - A) : t \in [0, 1] \,\}$ from $A$ to $M_A$. The medians from $B$ and from $C$ are defined in the same way, with the midpoints of $CA$ and of $AB$.

## Remarks

Some texts call the whole line through $A$ and $M_A$ the median.
The three medians meet at the [[centroid|0X7EEWES5G]]. See [[the medians meet at the centroid|0GPQ67XV7N]].
Each median divides the triangle into two triangles of equal area.
[[Apollonius's theorem|0NAKHQ9XYY]] gives the length of a median from the side lengths.

## Explanations

### L1 human

Join a corner of a triangle to the middle of the opposite side. The segment is a median. If the triangle is a thin plate of uniform material, it balances on each median.