Midpoint Formula
The midpoint formula finds the exact center point between two given points on the coordinate plane by averaging their coordinates.
Formula
M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)
Definition
The midpoint is the exact middle point of a line segment, found by averaging the x-coordinates and averaging the y-coordinates of the two endpoints: for $(x_1, y_1)$ and $(x_2, y_2)$, $M = ((x_1+x_2)/2, (y_1+y_2)/2)$, equidistant from both endpoints and equivalent to the arithmetic mean of the position vectors. More generally, the midpoint formula is the affine combination $(1/2)p + (1/2)q$ for points $p, q$ in $\mathbb{R}^n$, and the point dividing a segment $PQ$ in ratio $m:n$ is the weighted average $(nP + mQ)/(m+n)$; midpoints are preserved by affine transformations, which is why the centroid of a triangle (the intersection of medians, $G = (A + B + C)/3$) is an affine invariant.
Example
The midpoint of $(2, 4)$ and $(8, 10)$ is $x = (2+8)/2 = 5$, $y = (4+10)/2 = 7$, so $M = (5, 7)$, equidistant from both endpoints. If $M = (3, 6)$ is the midpoint and one endpoint is $A = (1, 2)$, the other endpoint is found by $x_B = 2(3) - 1 = 5$, $y_B = 2(6) - 2 = 10$, so $B = (5, 10)$, a reverse-midpoint technique that is tested frequently. Barycentric coordinates express any point $P$ in a triangle as a weighted average $P = \alpha A + \beta B + \gamma C$ where $\alpha + \beta + \gamma = 1$; the midpoint formula is simply the barycentric coordinate $(1/2, 1/2)$ of a segment.
Key Insight
Finding the midpoint is like averaging two numbers, done separately for x and y: it is always halfway between the two points in both directions at once, and is really a special case of a weighted average (the point $1/3$ of the way from $A$ to $B$ would be $A + (1/3)(B - A)$). Barycentric coordinates and the midpoint's role as an affine invariant are foundational ideas in computational geometry and finite element methods.