  The classical Ehresmann-Bruhat order describes the possible degenerations of
a pair of flags in a linear space V under linear transformations of V; or
equivalently, it describes the closure of an orbit of GL(V) acting diagonally
on the product of two flag varieties.
  We consider the degenerations of a triple consisting of two flags and a line,
or equivalently the closure of an orbit of GL(V) acting diagonally on the
product of two flag varieties and a projective space. We give a simple rank
criterion to decide whether one triple can degenerate to another. We also
classify the minimal degenerations, which involve not only reflections (i.e.,
transpositions) in the Weyl group S_n, n=dim(V), but also cycles of arbitrary
length. Our proofs use only elementary linear algebra and combinatorics.
