Equation for Line through Two Points in Complex Plane/Symmetric Form

Theorem

Let $z_1, z_2 \in \C$ be complex numbers.

Let $L$ be a straight line through $z_1$ and $z_2$ in the complex plane.

$L$ can be expressed by the equation:

$z = \dfrac {m z_1 + n z_2} {m + n}$

This form of $L$ is known as the symmetric form.

Proof

Let $z_1$ and $z_2$ be represented by the points $A = \tuple {x_1, y_1}$ and $B = \tuple {x_2, y_2}$ respectively in the complex plane.

Let $z$ be an arbitrary point on $L$ represented by the point $P$.

As $AP$ and $AB$ are collinear:

$m AP = n PB$

and so:

$m \paren {z - z_1} = n \paren {z_2 - z_1}$

The result follows.

$\blacksquare$

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Solved Problems: Graphical Representations of Complex Numbers. Vectors: $11$