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

A family of geometric operators on triangles with two complex variables