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General case

Cartesian plane. Line drawn through points A, B, P and Q.
Detailed description

We can find a formula for the midpoint of any interval. Suppose that \(P(x_1, y_1)\) and \(Q(x_2, y_2)\) are two points and let \(M(x, y)\) be the midpoint.

Triangles PMS and MQT are congruent triangles (AAS),
and so \(PS = MT\) and \(MS = QT\).

Hence the x-coordinate of M is the average of \(x_1\) and \(x_2\), and the y-coordinate of M is the average of \(y_1\) and \(y_2\). Therefore:

\(x = \dfrac{x_1 + x_2}{2} \text{and}\ y = \dfrac{y_1 + y_2}{2}\).