Geometric classification of triangulations and their enumeration in a convex polygon (Q1324413)

The authors consider triangulations of a convex \(n\)-gon in the plane. The length of a diagonal joining two vertices \(i\) and \(j\) of the \(n\)-gon is the number of edges in the shorter path from \(i\) to \(j\) through consecutive vertices of the \(n\)-gon. They show that any triangulation either contains a (bisector) diagonal of length \(n/2\) or it contains a unique (equilateral, isosceles; or scalene) triangular face with perimeter \(n\). The authors determine the number of non-isomorphic triangulations of these four types.