Some New Transversality Properties

The theorems of Ceva and Menelaus are concerned with cyclic products of ratios of lengths of collinear segments of triangles or more general polygons. These segments have one endpoint at a vertex of the polygon and one at the intersection point of a side with a suitable line. To these classical results we have recently added a ‘selftransversality theorem’ in which the ‘suitable line’ is determined by two other vertices. Here we present additional ‘transversality’ properties in which the ‘suitable line’ is determined either by a vertex and the intersection point of two diagonals, or by the intersection points of two pairs of such diagonals. Unexpectedly it turns out that besides several infinite families of systematic cases there are also a few ‘sporadic’ cases.