Area of a Triangle

For a triangle with vertices $(x_1,y_1),(x_2,y_2),(x_3,y_3)$, the familiar shoelace area formula

can be written compactly as the absolute value of a determinant:

The modulus (absolute value) is essential: the raw determinant can come out negative depending on the order (clockwise or anticlockwise) in which the vertices are listed, while area itself is always non-negative — the sign of the determinant is a labelling artefact, not a geometric fact.

Collinearity test. Three points are collinear (lie on a single straight line) exactly when the 'triangle' they would form has zero area:

This single determinant condition is often far quicker than comparing slopes pairwise, and — via the row operations of Property 7.3.2(8) — it frequently simplifies to almost nothing before it even needs expanding. …