Equation relating Points of Parallelogram in Complex Plane

Theorem
Let $ABVU$ be a parallelogram in the complex plane whose vertices correspond to the complex numbers $a, b, v, u$ respectively.

Let $\angle BAU = \alpha$.

Let $\cmod {UA} = \lambda \cmod {AB}$.


 * Parallelogram-in-Complex-Plane.png

Then:
 * $u = \paren {1 - q} a + q b$
 * $v = -q a + \paren {1 + q} b$

where:
 * $q = \lambda e^{i \alpha}$

Proof
From Geometrical Interpretation of Complex Subtraction, the four sides of $UABC$ can be defined as:

Thus:

Then we have: