Equation for Line through Two Points in Complex Plane/Formulation 1

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:

$\map \arg {\dfrac {z - z_1} {z_2 - z_1} } = 0$

Proof

Let $z$ be a point on the $L$.

Then:

$z - z_1 = b \paren {z - z_2}$

where $b$ is some real number.

Then:

\(\ds b\)

\(=\)

\(\ds \frac {z - z_1} {z - z_2}\)

\(\ds \leadsto \ \ \)

\(\ds \map \arg {\frac {z - z_1} {z_2 - z_1} }\)

\(=\)

\(\ds \arg b\)

\(\ds \)

\(=\)

\(\ds 0\)

as $b$ is real

$\blacksquare$

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Miscellaneous Problems: $137$