Determinant formula for triangle area ($\mathrm{area}(p,q,r)=\frac{1}{2}|\det (q-p,r-p)|$)

Two side vectors span a parallelogram whose signed area is their determinant. Its two congruent triangles give the factor one-half, and the absolute value removes orientation. Reversing the vertex order changes the sign but not the area; collinear vertices give zero.