Area of a Triangle using Determinants

The area of a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), \((x_3, y_3)\) is

\[\Delta = \frac12\left|\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}\right|.\]

Because area is positive, we take the absolute value. When the area is given and a coordinate is unknown, set the determinant equal to \(+2\Delta\) and \(-2\Delta\) – there are usually two answers.

Collinearity

Three points are collinear exactly when this determinant is 0 (the “triangle” has no area). This also gives the equation of a line through two points: \((x, y)\) lies on the line through \((x_1, y_1)\) and \((x_2, y_2)\) iff the determinant is 0.