$Midpoint$

Midpoint

Find the midpoint of a segment, the point dividing a segment in a given ratio and the endpoint from the midpoint in 2D and 3D.

What is known:

Midpoint of segment AB

Point dividing AB in the ratio λ = AM : MB

Endpoint B from start A and midpoint M

Space: in the plane (x, y) in space (x, y, z)

Coordinates are whole numbers, decimals or fractions: 3, −1.5, 2/3.

Test yourself

The midpoint calculator finds the coordinates of the midpoint of a segment from its endpoints, the point that divides a segment in a given ratio, and the second endpoint from one endpoint and the midpoint — in the plane and in space.

Formulas

Problem	Formula
Midpoint of AB	x_M = (x_A + x_B)/2, y_M = (y_A + y_B)/2, z_M = (z_A + z_B)/2
Division in the ratio λ = AM : MB	x_M = (x_A + λx_B)/(1 + λ), the same for y, z
Endpoint B from A and midpoint M	x_B = 2x_M − x_A, the same for y, z

Example 1. A(2, −4, 1), B(6, 8, −3).
M((2 + 6)/2, (−4 + 8)/2, (1 − 3)/2) = M(4, 2, −1).

Example 2. Divide the segment A(1, 2, 0), B(7, 11, 6) in the ratio 2 : 1.
M((1 + 14)/3, (2 + 22)/3, (0 + 12)/3) = M(5, 8, 4).

The midpoint is the special case λ = 1. The centroid of a triangle divides each median in the ratio 2 : 1 from the vertex; its coordinates are the averages of the coordinates of the vertices.