Midpoint Formula Calculator

The midpoint formula finds the point exactly halfway between two given points. It's just the average of the coordinates:

2D form — for points $(x_1, y_1)$ and $(x_2, y_2)$:

$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

3D form — for points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$:

$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right)$

Why averaging works: the midpoint divides the segment in a $1:1$ ratio, and the coordinates of any point on the segment are weighted averages of the endpoints. With equal weights ($1/2$ each), you get the simple arithmetic mean.

The midpoint formula appears constantly in coordinate geometry: finding the center of a circle from its diameter, the centroid of a triangle, parallelograms, perpendicular bisectors, and any problem involving 'halfway between'.

Step-by-Step

1. Identify the two points $(x_1, y_1)$ and $(x_2, y_2)$.
2. Average the x-coordinates: $\frac{x_1 + x_2}{2}$.
3. Average the y-coordinates: $\frac{y_1 + y_2}{2}$.
4. Combine into the midpoint $(M_x, M_y)$.

No subtraction, no squares, no roots — much simpler than the distance formula.

Reverse Problem: Find Endpoint from Midpoint

If $M = (M_x, M_y)$ is the midpoint of $(x_1, y_1)$ and $(x_2, y_2)$, you can solve for either endpoint:

$x_2 = 2 M_x - x_1, \quad y_2 = 2 M_y - y_1$

Double the midpoint, subtract the known endpoint.

Generalization: Section Formula

For a point dividing a segment in ratio $m : n$ (not just $1:1$):

$P = \left(\frac{m x_2 + n x_1}{m + n}, \frac{m y_2 + n y_1}{m + n}\right)$

The midpoint formula is the special case $m = n = 1$.

Geometric Applications

• Center of a circle from diameter endpoints: just the midpoint.
• Centroid of a triangle: average of all three vertex coordinates (generalizes midpoint to 3 points).
• Perpendicular bisector: a line through the midpoint perpendicular to the original segment.
• Diagonals of a parallelogram: the midpoints of both diagonals coincide — useful for proving a quadrilateral is a parallelogram.

• Subtracting instead of adding: midpoint averages — $\frac{x_1 + x_2}{2}$, not $\frac{x_2 - x_1}{2}$. Subtraction belongs to the distance formula.
• Forgetting to divide each coordinate: the divisor 2 applies separately to the x-sum and the y-sum. It's not a single division at the end.
• Sign errors with negative coordinates: $\frac{-3 + 7}{2} = 2$, not $-2$ or $5$. Add carefully.
• Mixing midpoint and slope formulas: midpoint averages, slope subtracts. They look similar but answer different questions.
• Forgetting to update for 3D: if your problem is in 3D, include the z-average. If 2D, don't add a phantom z.