Area of Triangle in Determinant Form/Proof 1

Proof

 * AreaOfTriangleComplex.png

Let $A$, $B$ and $C$ be defined as complex numbers in the complex plane.

The vectors from $C$ to $A$ and from $C$ to $B$ are given by:


 * $z_1 = \paren {x_1 - x_3} + i \paren {y_1 - y_3}$
 * $z_2 = \paren {x_2 - x_3} + i \paren {y_2 - y_3}$

From Area of Triangle in Terms of Side and Altitude, $\AA$ is half that of a parallelogram contained by $z_1$ and $z_2$.

Thus: