---
tag: 06F3Y6CMG9
kind: definition
name: Midpoint
msc: [51N10]
defines: 0S706JBZBG
uses: [0KJH6SRFZK]
sources:
  - url: https://en.wikipedia.org/wiki/Midpoint
    locator: Formula
---

## Definition

Let $K$ be a field in which $2 \neq 0$, and let $A$ be an [[affine space|09XNAPTJQ7]] over a vector space $V$ over $K$. Let $p$ and $q$ be points of $A$.
The midpoint of $p$ and $q$ is the point
$$\operatorname{midpoint}(p, q) = p + \tfrac{1}{2}(q - p).$$
It is the [[affine combination|0KJH6SRFZK]] $\frac{1}{2} p + \frac{1}{2} q$.

## Remarks

In $\mathbb{R}^n$, the midpoint of $(p_1, \dots, p_n)$ and $(q_1, \dots, q_n)$ is $\left(\frac{p_1 + q_1}{2}, \dots, \frac{p_n + q_n}{2}\right)$.
In a Euclidean space, the midpoint $m$ of $p$ and $q$ satisfies $d(p, m) = d(m, q) = \frac{1}{2} d(p, q)$. It is the only point with this property.
The midpoint is symmetric: $\operatorname{midpoint}(p, q) = \operatorname{midpoint}(q, p)$. Also $\operatorname{midpoint}(p, p) = p$.

## Explanations

### L1 human

The midpoint is the average of the two points. To find it, take the average of each coordinate. For example, the midpoint of $(0, 0)$ and $(4, 6)$ is $(2, 3)$.