KöMaL 2025 B5493

Show a function $f$ that assigns a non-negative value to every vector in the plane satisfying the following: for an arbitrary triangle $ABC$ with vertices on hyperbola $x^2-y^2=1$ the area of the triangle equals $f{(\overrightarrow{AB})\cdot f(\overrightarrow{BC})\cdot f(\overrightarrow{CA})}$.