Midpoint

Definition

Let $K$ be a field in which $2 \neq 0$, and let $A$ be an affine space 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 $\frac{1}{2} p + \frac{1}{2} q$.

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 $(2, 3)$.