Area of Triangle in Determinant Form/Proof 2

Proof

 * Area-of-Triangle-Determinant.png

Let $A$, $B$ and $C$ be as defined..

Let $O$ be the origin of the Cartesian plane in which $\triangle ABC$ is embedded.

Taking into account the signs of the areas of the various triangles involved:


 * $\triangle ABC = \triangle OAB + \triangle OBC + \triangle OCA$

as it is seen that $\triangle OBC$ and $\triangle OCA$ are described in clockwise sense.

From proof 2 of Area of Triangle in Determinant Form with Vertex at Origin:

The result follows.