Midpoint Formula

Definition

The Midpoint Formula calculates the coordinates of the exact center point between two endpoints. It represents the arithmetic mean of the coordinates of the points.

For 2D points $(x_1, y_1)$ and $(x_2, y_2)$: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

For 3D points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right)$

• How to read: “The midpoint M equals the point whose coordinates are the average of the x-coordinates, the average of the y-coordinates, and (in 3D) the average of the z-coordinates.”
• Meaning: M is the unique point that divides a line segment into two congruent, equal parts.

Why It Matters

The midpoint represents the fundamental concepts of balance, symmetry, and averaging. In structural engineering, it identifies the balance point of beams; in computer graphics (LERP), it is the base case for interpolation; and in geometry, it is the key to proving theorems about bisectors and medians.

Core Concepts

• Arithmetic Mean: The formula treats each dimension independently, calculating the average position along each axis.
• Equidistance: The midpoint $M$ is guaranteed to be equidistant from both endpoints ($P_1M = MP_2$) and collinear with them.
• Vector Representation: Using position vectors $\mathbf{r_1}$ and $\mathbf{r_2}$, the midpoint vector $\mathbf{m}$ is: $\mathbf{m} = \frac{1}{2}(\mathbf{r_1} + \mathbf{r_2})$
• Section Formula (Generalization): The midpoint is a special case of the section formula (weighted average) where the ratio is $1:1$. For a point dividing a segment at fraction $t$: $\mathbf{P} = (1-t)\mathbf{r_1} + t\mathbf{r_2}$ How to read: “The point $P$ equals one minus $t$ times vector $r$ sub one plus $t$ times vector $r$ sub two.” Meaning: This is linear interpolation (LERP), where $t=0.5$ yields the midpoint.

Connected Concepts

• Distance Formula
• Vectors in Space
• Centroid of a Triangle
• Analytic Proof Strategies