Triangle Inequality/Complex Numbers/Proof 3

Proof
Let $z_1$ and $z_2$ be represented by the points $A$ and $B$ respectively in the complex plane.

From Geometrical Interpretation of Complex Addition, we can construct the parallelogram $OACB$ where:
 * $OA$ and $OB$ represent $z_1$ and $z_2$ respectively
 * $OC$ represents $z_1 + z_2$.


 * Triangle-Inequality-Complex.png

As $OACB$ is a parallelogram, we have that $OB = AC$.

The lengths of $OA$, $AC$ and $OC$ are:

But $OA$, $OB$ and $OC$ form the sides of a triangle.

The result then follows directly from Sum of Two Sides of Triangle Greater than Third Side.