  In the projective plane, we consider congruences of straight lines with the
combinatorics of the square grid and with all elementary quadrilaterals
possessing touching inscribed conics. The inscribed conics of two
combinatorially neighbouring quadrilaterals have the same touching point on
their common edge-line. We suggest that these nets are a natural projective
generalisation of incircular nets. It is shown that these nets are planar
Koenigs nets. Moreover, we show that general Koenigs nets are characterised by
the existence of a 1-parameter family of touching inscribed conics. It is shown
that the lines of any grid of quadrilaterals with touching inscribed conics are
tangent to a common conic. These grids can be constructed via polygonal chains
that are inscribed in conics. The special case of billiards in conics
corresponds to incircular nets.
