Three Conics Determine a Cubic

Given a cubic K in the real projective plane. Then for each point P there is a conic C_P C P associated to P . The conic C_P C P is called the polar conic of K with respect to the pole P . We investigate the situation when three conics C₁ C 1 , C₂ C 2 , and C₃ C 3 are polar conics of K with respect to the poles P₁ P 1 , P₂ P 2 , and P₃ P 3 , respectively. In particular, we give an elementary proof—without using any results from algebraic geometry—that any three conics C₁ C 1 , C₂ C 2 , C₃ C 3 in general position, satisfying only a non-degeneracy condition, determine a unique cubic K and three points P₁ P 1 , P₂ P 2 , P₃ P 3 , such that C₁ C 1 , C₂ C 2 , C₃ C 3 are polar conics of K with respect to the three poles P₁ P 1 , P₂ P 2 , P₃ P 3 . This can be seen as a higher degree variant of von Staudt’s Theorem.