Equation of Straight Line in Plane/Two-Point Form/Proof 4

Proof

 * Straight-line-2-points-form-Proof-4.png

Let $z_1 = x_1 + i y_1$ and $z_2 = x_2 + i y_2$ be the position vectors of the points $A$ and $B$ embedded in the complex plane.

Let $z = x + i y$ be the position vector of an arbitrary point $P$ on the straight line $AB$.

From the diagram:

and:

Then: