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We consider congruences of straight lines in a plane with the combinatoricsof the square grid, with all elementary quadrilaterals possessing an incircle.It is shown that all the vertices of such nets (we call them incircular orIC-nets) lie on confocal conics.  Our main new results are on checkerboard IC-nets in the plane. These arecongruences of straight lines in the plane with the combinatorics of the squaregrid, combinatorially colored as a checkerboard, such that all black coordinatequadrilaterals possess inscribed circles. We show how this larger class ofIC-nets appears quite naturally in Laguerre geometry of oriented planes andspheres, and leads to new remarkable incidence theorems. Most of our resultsare valid in hyperbolic and spherical geometries as well. We present alsogeneralizations in spaces of higher dimension, called checkerboard IS-nets. Theconstruction of these nets is based on a new 9 inspheres incidence theorem.

Incircular nets and confocal conics