  Given a plane triangle $\Delta$, one can construct a new triangle $\Delta'$
whose vertices are intersections of two cevian triples of $\Delta$. We extend
the family of operators $\Delta\mapsto\Delta'$ by complexifying the defining
two cevian parameters and study its rich structure from arithmetic-geometric
viewpoints. We also find another useful parametrization of the operator family
via finite Fourier analysis and apply it to investigate area-preserving
operators on triangles.
