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 \(C_1\) $C_1$ , \(C_2\) $C_2$ , and \(C_3\) $C_3$ are polar conics of K with respect to the poles \(P_1\) $P_1$ , \(P_2\) $P_2$ , and \(P_3\) $P_3$ , respectively. In particular, we give an elementary proof—without using any results from algebraic geometry—that any three conics \(C_1\) $C_1$ , \(C_2\) $C_2$ , \(C_3\) $C_3$ in general position, satisfying only a non-degeneracy condition, determine a unique cubic K and three points \(P_1\) $P_1$ , \(P_2\) $P_2$ , \(P_3\) $P_3$ , such that \(C_1\) $C_1$ , \(C_2\) $C_2$ , \(C_3\) $C_3$ are polar conics of K with respect to the three poles \(P_1\) $P_1$ , \(P_2\) $P_2$ , \(P_3\) $P_3$ . This can be seen as a higher degree variant of von Staudt’s Theorem.