Polygons in Euclidean Buildings of rank 2

Let X be a symmetric space of noncompact type (e.g. SL(n)/SO(n)) or a thick Euclidean bulding. We will discuss following geometric question: which are the possible side lengths of polygons in X?
The full invariant of an oriented geodesic segment in a symmetric space X=G/K modulo the action of G is a vector in \R^{rank(X)}. Thus in our context the appropriate notion of length is given by this vector. The same notion of vector valued length can be defined for Euclidean buildings.
A special case of the geometric question above is closely related to the Eigenvalue Problem: How are the eigenvalues of two Hermitian matrices related to the eigenvalues of their sum?