Those Amazing Desargues Configurations

Abstract

Given 2 triangles in a plane over a field $F$ which are in perspective from a vertex $V$, the resulting Desargues line or axis $l$ may or may not be on $V$. To avoid degenerate cases, we assume that the union of the vertices of the 2 triangles is a set of six points with no three collinear. Our work then provides a detailed analysis of situations when $V$ is on $l$ for any $F$, finite or infinite.