Triangle Inequality/Complex Numbers/Examples/3 Arguments/Proof 3

Proof
Let $z_1$, $z_2$ and $z_3$ be represented by the points $A$, $B$ and $C$ respectively in the complex plane.

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

Also from Geometrical Interpretation of Complex Addition, we can construct the parallelogram $OCED$ where:
 * $OC$ and $OD$ represent $z_3$ and $z_1 + z_2$ respectively
 * $OD$ represents $z_1 + z_2 + z_3$.


 * Triangle-Inequality-Complex-3-Arguments.png

As $OADB$ is a parallelogram, we have that $OB = AD$.

The lengths of $OA$, $AD$ and $OD$ are:

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

Thus from Sum of Two Sides of Triangle Greater than Third Side:
 * $\cmod {z_1 + z_2} \le \cmod {z_1} + \cmod {z_2}$

Similarly, as $OCED$ is a parallelogram, we have that $OD = CE$.

The lengths of $OC$, $CE$ and $OE$ are:

But $OC$, $CE$ and $OE$ form the sides of a triangle.

Thus from Sum of Two Sides of Triangle Greater than Third Side:
 * $\cmod {z_1 + z_2 + z_3} \le \cmod {z_1 + z_2} + \cmod {z_3}$

The result follows.