The ternary rings of Desarguesian and Pappian planes - a simple proof using perspectivities (Q1059849)

A Pappian plane has the following property. If \(\phi\) is a projectivity of a line \(\ell\) onto another line m such that \((\ell \cap m)\phi =\ell \cap m\) then \(\phi\) is a perspectivity. - Conversely, if a projective plane \(\pi\) satisfies this condition then \(\pi\) is Pappian. Using this well-known fact the author gives a direct, simple proof of the following classical theorem. The ternary ring of a Pappian projective plane is a ternary ring over a commutative field. Simultaneously, the proof yields that the ternary ring of a Desarguesian plane is a linear one over a skew field.