Fuzzy plane geometry: triangles

A fuzzy triangle T (with a discrete-valued membership function) can be regarded as a nest of parallel-sided triangles T_i with successively higher membership values. Such a nest is determined by its max projections on any two of its “sides”. The area (perimeter) of T is a weighted sum of the areas (perimeters) of the T _i´s. The side lengths and altitudes of T can also be defined as weighted sums obtained from projections; using these definitions, the perimeter of T is the sum of the side lengths, and the side lengths are related to the vertex angles by the Law of Sines, but there is no simple relationship between the area of T and the products of the side lengths and altitudes