A generalized conjecture about cutting a polygon into triangles of equal areas
It is well known that a square cannot be dissected into an odd number of triangles having equal areas, and the same statement holds true for centrally symmetric polygons. In view of this question, also polygons were considered whose edges are parallel to the \(x\)- or \(y\)-axes, and polygons obtained from the square by altering opposite edges in a way such that the resulting polygon still tiles the plane via integer translation vectors. All these are special cases of so-called special polygons introduced in the present paper: Let \(P\) be a simply connected polygon with oriented boundary. Orienting the edges of \(P\) in the sense of this orientation and calling two edges equivalent if they are parallel, then \(P\) is a special polygon if the sum of the respective vectors in each equivalence class is 0. The author poses the conjecture that any special polygon cannot be cut into an odd number of equiareal triangles, and he confirms this conjecture for polygons with at most six sides.
scientific article; zbMATH DE number 1495785