  This paper studies the discrete differential geometry of the checkerboard
pattern inscribed in a quadrilateral net by connecting edge midpoints. It turns
out to be a versatile tool which allows us to consistently define principal
nets, Koenigs nets and eventually isothermic nets as a combination of both.
  Principal nets are based on the notions of orthogonality and conjugacy and
can be identified with sphere congruences that are entities of Moebius
geometry. Discrete Koenigs nets are defined via the existence of the so called
conic of Koenigs. We find several interesting properties of Koenigs nets,
including their being dualizable and having equal Laplace invariants.
  Koenigs nets that are also principal are defined as isothermic nets. We prove
that the class of isothermic nets is invariant under both dualization and
Moebius transformations. Among other things, this allows a natural construction
of discrete minimal surfaces and their Goursat transformations.
